{ "format": "stapler-pack", "version": 1, "kind": "library", "meta": { "title": "Circle Geometry", "author": "Staples Education", "description": "A complete circle-geometry unit in six short, assignable quizzes: arcs and chords, circumference and area, equations of circles, inscribed angles, secant and tangent angles, and segment lengths. 35 procedural questions plus 15 that ask students to find the error, name the theorem, or identify what is missing.", "created": "2026-08-09T05:30:56.032Z", "license": "Proprietary - Staples Education" }, "payload": { "library": { "name": "Circle Geometry", "categories": [], "quizzes": [ { "id": "circles-arcs-chords", "title": "Arcs, Chords & Central Angles", "blurb": "Central angles and the arcs they cut off, congruent chords, and the perpendicular from the centre.", "meta": "10 questions · fluency + reasoning", "categories": [ "circles" ], "questions": [ { "id": "cg-001", "topic": "Central Angles & Arcs", "difficulty": "easy", "stem": "In circle O, central angle ∠AOB measures 74°. What is the measure of minor arc AB?", "choices": [ "37°", "74°", "148°", "286°" ], "correctIndex": 1, "explanations": [ "37° is half of 74°, which is the inscribed-angle relationship — an angle with its vertex ON the circle intercepts an arc twice its size. This vertex is at the centre, where no halving happens.", "Correct. A central angle has its vertex at the centre, and its intercepted arc has exactly the same measure as the angle.", "148° doubles the angle. Doubling is what you do to an INSCRIBED angle to get its arc; a central angle already equals its arc.", "286° is the MAJOR arc AB, the long way round (360° − 74°). The question asks for the minor arc." ], "hint": "Where is the vertex of the angle — at the centre, or on the circle? That decides whether any halving or doubling applies.", "solution": [ "A central angle has its vertex at the centre of the circle.", "Its intercepted arc has exactly the same measure as the angle.", "arc AB = 74°" ], "diagram": "OAB74°", "diagramCaption": "Not drawn to scale" }, { "id": "cg-002", "topic": "Central Angles & Arcs", "difficulty": "easy", "stem": "Minor arc CD in circle P measures 128°. What is the measure of central angle ∠CPD?", "choices": [ "64°", "128°", "232°", "256°" ], "correctIndex": 1, "explanations": [ "64° halves the arc. That relationship belongs to inscribed angles, whose vertex sits on the circle — not to a central angle.", "Correct. Central angle and intercepted arc are equal in measure, so the angle is 128°.", "232° is 360° − 128°, the reflex angle at P that opens onto the major arc. ∠CPD is the ordinary angle intercepting the minor arc.", "256° doubles the arc. Nothing in the central-angle relationship doubles; the two measures are simply equal." ], "solution": [ "Central angle and intercepted arc are equal in measure.", "∠CPD = 128°" ] }, { "id": "cg-003", "topic": "Central Angles & Arcs", "difficulty": "medium", "stem": "Points A, B and C lie on circle O. Arc AB = 112° and arc BC = 145°. What is the measure of arc CA (the remaining arc)?", "choices": [ "93°", "103°", "115°", "257°" ], "correctIndex": 1, "explanations": [ "93° would make the three arcs total 350°. Check the sum: the arcs of a full circle must come to exactly 360°.", "Correct. The three arcs make a complete circle, so arc CA = 360° − 112° − 145° = 103°.", "115° totals 372°, overshooting the full circle by 12°.", "257° is 112° + 145°, the sum of the two GIVEN arcs rather than what is left over. Subtract that sum from 360° instead." ], "hint": "Three arcs with no gaps and no overlaps make one complete trip around the circle.", "solution": [ "The three arcs together make one complete circle, so they sum to 360°.", "arc CA = 360° − 112° − 145°", "arc CA = 103°" ], "diagram": "ABC112°145°?" }, { "id": "cg-004", "topic": "Central Angles & Arcs", "difficulty": "medium", "stem": "AC is a diameter of circle O, and B is a point on the circle. If arc AB = 63°, what is the measure of arc BC?", "choices": [ "27°", "63°", "117°", "297°" ], "correctIndex": 2, "explanations": [ "27° is 90° − 63°. Complementary angles are not involved here — a diameter cuts off a semicircle of 180°, not a right angle of 90°.", "63° would make arc AB and arc BC equal, which only happens if B is the midpoint of the semicircle. Nothing here says it is.", "Correct. A diameter splits the circle into two semicircles of 180° each, so arc BC = 180° − 63° = 117°.", "297° is 360° − 63°, which is the major arc from A back to B the long way. B and C are both on the same semicircle as measured here." ], "hint": "What is the arc measure of a semicircle?", "solution": [ "AC is a diameter, so each side of it is a semicircle of 180°.", "arc BC = 180° − 63°", "arc BC = 117°" ] }, { "id": "cg-005", "topic": "Arcs & Chords", "difficulty": "easy", "stem": "In circle O, chord AB is congruent to chord CD. If arc AB = 82°, what is the measure of arc CD?", "choices": [ "41°", "82°", "98°", "278°" ], "correctIndex": 1, "explanations": [ "41° halves the arc. There is no halving in the chord–arc relationship; congruent chords cut off arcs of the same size.", "Correct. In the same circle, congruent chords intercept congruent arcs — so arc CD is also 82°.", "98° is the supplement of 82°. Supplements appear with cyclic quadrilaterals, not with congruent chords.", "278° is the major arc (360° − 82°). Both chords cut off minor arcs of equal measure." ], "hint": "Equal chords sit the same distance from the centre, so they subtend equal central angles.", "solution": [ "In the same circle, congruent chords intercept congruent arcs.", "arc CD = arc AB = 82°" ] }, { "id": "cg-006", "topic": "Arcs & Chords", "difficulty": "medium", "stem": "A chord of length 24 is drawn in a circle of radius 13. How far is the chord from the centre of the circle?", "choices": [ "5", "6.5", "7", "12" ], "correctIndex": 0, "explanations": [ "Correct. The perpendicular from the centre bisects the chord, giving a right triangle with legs 12 and d and hypotenuse 13. Then d = √(169 − 144) = 5.", "6.5 is half the radius. Halving the radius has no geometric meaning here — the distance comes out of the Pythagorean relationship, not out of the radius alone.", "7 is 13 − 6, mixing the radius with part of the chord by subtracting. The three lengths form a right triangle, so they combine by a² + b² = c², not by subtraction.", "12 is half the chord, which is one LEG of the right triangle. The question asks for the other leg." ], "hint": "Drop a perpendicular from the centre to the chord. What does it do to the chord, and what triangle appears?", "solution": [ "Drop a perpendicular from the centre to the chord; it bisects the chord into halves of 12.", "That half-chord, the distance d and the radius form a right triangle: 12² + d² = 13².", "d = √(169 − 144) = √25", "d = 5" ], "diagram": "ABO?1312", "diagramCaption": "The segment from the centre meets the chord at a right angle" }, { "id": "cg-007", "topic": "Arcs & Chords", "difficulty": "easy", "stem": "In circle O, diameter EF is perpendicular to chord GH, meeting it at point M. If GH = 30, what is the length of GM?", "choices": [ "7.5", "15", "30", "60" ], "correctIndex": 1, "explanations": [ "7.5 is a quarter of the chord. The perpendicular cuts the chord into two equal halves, not four quarters.", "Correct. A diameter perpendicular to a chord bisects it, so GM = 30 ÷ 2 = 15.", "30 is the whole chord GH. GM is only the piece from G to the point where the diameter crosses.", "60 doubles the chord. Nothing here doubles — the diameter divides the chord in two." ], "solution": [ "A diameter perpendicular to a chord bisects that chord.", "GM = 30 ÷ 2", "GM = 15" ] }, { "id": "cg-008", "topic": "Arcs & Chords", "difficulty": "hard", "stem": "In a circle of radius 25, two chords are each 7 units from the centre. What is the length of each chord?", "choices": [ "24", "32", "48", "96" ], "correctIndex": 2, "explanations": [ "24 is √(625 − 49) rounded incorrectly, and in any case that value is HALF the chord. √576 = 24, so the full chord is twice that.", "32 does not come out of the Pythagorean relationship here; check √(25² − 7²) carefully.", "Correct. Half the chord is √(25² − 7²) = √576 = 24, so the full chord is 48. Chords equidistant from the centre are congruent, so both measure 48.", "96 doubles the answer a second time. The half-chord is 24, so one doubling gives the full chord." ], "hint": "Find the half-chord with the Pythagorean theorem first, then remember which length the question is asking for.", "solution": [ "Half the chord, the distance 7 and the radius 25 form a right triangle.", "half = √(25² − 7²) = √(625 − 49) = √576 = 24", "chord = 2 × 24", "chord = 48" ] }, { "id": "cg-041", "topic": "Reasoning: What's Missing", "difficulty": "hard", "stem": "A chord of a circle measures 16. A student is asked to find the chord's distance from the centre and says it cannot be done. What single additional piece of information would make it solvable?", "choices": [ "The length of a second chord", "The radius of the circle", "Nothing else is needed; the answer is 8", "The measure of the arc the chord cuts off" ], "correctIndex": 1, "explanations": [ "A second chord gives no new information about this one unless its distance from the centre is also known.", "Correct. With the radius r, the perpendicular from the centre bisects the chord into halves of 8, and the distance is √(r² − 64). Without r there is nothing to compute.", "8 is HALF THE CHORD, not the distance to the centre. Those are two different legs of the same right triangle.", "The arc measure would fix the shape but not the size — the same arc measure occurs in circles of every size, each giving a different distance." ], "hint": "Draw the right triangle. Which of its three sides do you already know?", "solution": [ "The perpendicular from the centre bisects the chord, giving two halves of 8.", "That half-chord and the distance d are the legs of a right triangle whose hypotenuse is the radius.", "d = √(r² − 8²) = √(r² − 64), which cannot be evaluated without r.", "The missing piece is the radius of the circle." ], "kind": "reasoning" }, { "id": "cg-044", "topic": "Reasoning: Choose the Theorem", "difficulty": "medium", "stem": "A carpenter needs to find the exact centre of a circular tabletop. Which construction locates it, and why does it work?", "choices": [ "Draw a tangent and measure inward by the radius.", "Draw one chord and take its midpoint; the centre is always the midpoint of a chord.", "Draw two non-parallel chords and construct each one's perpendicular bisector; the centre is where they meet.", "Draw the longest chord you can find and take one endpoint." ], "correctIndex": 2, "explanations": [ "This assumes the radius is already known, which it is not — and the direction to measure would still be unknown.", "The centre is the midpoint of a chord only when that chord is a diameter — and finding a diameter is essentially the problem being solved.", "Correct. The perpendicular bisector of any chord passes through the centre, so two of them (from non-parallel chords) intersect at exactly one point: the centre.", "An endpoint of a chord lies on the circle, not at the centre. The longest chord IS a diameter, but its endpoints are as far from the centre as possible." ], "hint": "Which line related to a chord is guaranteed to pass through the centre?", "solution": [ "The perpendicular bisector of any chord passes through the centre of the circle.", "Draw two chords that are not parallel, and construct each one's perpendicular bisector.", "Two such bisectors meet at exactly one point, and that point is the centre." ], "kind": "reasoning" } ] }, { "id": "circles-circumference-area", "title": "Circumference, Area & Sectors", "blurb": "Circumference and area from a radius or a diameter, plus arc length and sector area.", "meta": "6 questions · fluency + reasoning", "categories": [ "circles", "area-perimeter" ], "questions": [ { "id": "cg-009", "topic": "Circumference & Area", "difficulty": "easy", "stem": "A circle has radius 7 cm. What is its circumference, in terms of π?", "choices": [ "28π cm", "7π cm", "49π cm", "14π cm" ], "correctIndex": 3, "explanations": [ "28π uses 2πd rather than 2πr — it doubles the diameter instead of the radius, giving twice the correct answer.", "7π uses C = πr, which is missing the factor of 2. Circumference is π times the DIAMETER, and the diameter is twice the radius.", "49π is the AREA (πr² = π·49). Area is measured in square units; circumference is a length.", "Correct. C = 2πr = 2π(7) = 14π cm." ], "hint": "Circumference is π times the diameter. What is the diameter here?", "solution": [ "C = 2πr", "C = 2π(7)", "C = 14π cm" ] }, { "id": "cg-010", "topic": "Circumference & Area", "difficulty": "easy", "stem": "A circle has diameter 18 in. What is its area, in terms of π?", "choices": [ "18π in²", "36π in²", "81π in²", "324π in²" ], "correctIndex": 2, "explanations": [ "18π is π times the diameter, which is the circumference — a length, not an area.", "36π is 2π(18), doubling the diameter. That is not a formula for either quantity here.", "Correct. The radius is 18 ÷ 2 = 9, so A = πr² = π(9²) = 81π in².", "324π squares the DIAMETER instead of the radius. The formula needs r², and r is half of 18." ], "hint": "The area formula takes the radius, and you have been given the diameter.", "solution": [ "The radius is half the diameter: r = 18 ÷ 2 = 9.", "A = πr² = π(9²)", "A = 81π in²" ] }, { "id": "cg-011", "topic": "Circumference & Area", "difficulty": "medium", "stem": "In a circle of radius 10, an arc is intercepted by a central angle of 72°. What is the length of that arc, in terms of π?", "choices": [ "2π", "4π", "20π", "72π" ], "correctIndex": 1, "explanations": [ "2π halves the correct answer — check the fraction: 72/360 simplifies to 1/5, not 1/10.", "Correct. The arc is 72/360 = 1/5 of the circle, and the full circumference is 2π(10) = 20π. One fifth of 20π is 4π.", "20π is the ENTIRE circumference. The arc is only the fraction of it cut off by a 72° angle.", "72π multiplies by the angle instead of taking the angle as a fraction of 360°." ], "hint": "What fraction of the whole circle does 72° represent?", "solution": [ "The arc is 72/360 = 1/5 of the way around the circle.", "The whole circumference is C = 2π(10) = 20π.", "arc length = (1/5)(20π)", "arc length = 4π" ] }, { "id": "cg-012", "topic": "Circumference & Area", "difficulty": "hard", "stem": "A sector of a circle of radius 6 has a central angle of 120°. What is the area of the sector, in terms of π?", "choices": [ "4π", "12π", "24π", "36π" ], "correctIndex": 1, "explanations": [ "4π is 120/360 of the CIRCUMFERENCE (12π), not of the area. That fraction of the circumference gives the arc length, which is a length rather than an area.", "Correct. The sector is 120/360 = 1/3 of the circle, and the full area is π(6²) = 36π. One third of 36π is 12π.", "24π is two thirds of the circle's area — the fraction has been taken the wrong way round (240/360 rather than 120/360).", "36π is the area of the WHOLE circle. The sector is only the slice cut off by the 120° angle." ], "hint": "Take the same fraction of the area that the angle is of 360°.", "solution": [ "The sector is 120/360 = 1/3 of the circle.", "The whole area is A = π(6²) = 36π.", "sector area = (1/3)(36π)", "sector area = 12π" ] }, { "id": "cg-046", "topic": "Reasoning: Error Analysis", "difficulty": "easy", "stem": "A circle has diameter 20. A student computes the circumference as C = 2π(20) = 40π. What went wrong?", "choices": [ "They used the diameter in place of the radius in C = 2πr. The circumference is 20π.", "Nothing; 40π is correct.", "They should have used C = πr².", "They should have divided by 2 at the end, giving 10π." ], "correctIndex": 0, "explanations": [ "Correct. C = 2πr needs the RADIUS, which is 10. Either compute 2π(10) or use C = πd = π(20) — both give 20π.", "40π is twice the correct value, because the diameter was substituted where the radius belongs.", "πr² is the AREA formula, and it produces square units rather than a length.", "Dividing 40π by 2 does reach 20π, but by accident — the fix is to substitute the radius, not to patch the result." ], "hint": "Write down which quantity each formula expects before substituting.", "solution": [ "C = 2πr takes the RADIUS, and the student used the diameter.", "The radius is 20 ÷ 2 = 10.", "C = 2π(10) = 20π, which C = πd = π(20) confirms." ], "kind": "reasoning" }, { "id": "cg-047", "topic": "Reasoning: What's Missing", "difficulty": "medium", "stem": "A student is asked for the AREA of a sector with central angle 60° in a circle of radius 12, and answers \"4π, because 60/360 of the circumference 24π is 4π.\" What is the mistake?", "choices": [ "They found the arc LENGTH rather than the sector area; the area is 24π.", "The fraction should be 60/180.", "Nothing is wrong; 4π is the area.", "They should have used the diameter instead of the radius." ], "correctIndex": 0, "explanations": [ "Correct. Taking that fraction of the circumference gives the arc length. For AREA, take the same fraction of the circle's area: ⅙ × π(12²) = ⅙ × 144π = 24π.", "60/360 is the right fraction — a sector's share of the circle is its angle over 360°.", "4π is a length, not an area, and the units alone give it away.", "The radius was used correctly. The error was taking a fraction of the wrong total." ], "hint": "The fraction was right. Check what it was a fraction OF.", "solution": [ "Taking a fraction of the CIRCUMFERENCE gives an arc length, not an area.", "For area, take the same fraction of the circle's AREA.", "The fraction is 60/360 = 1/6, and the circle's area is π(12²) = 144π.", "sector area = (1/6)(144π) = 24π" ], "kind": "reasoning" } ] }, { "id": "circles-equations", "title": "Equations of Circles", "blurb": "Standard form, reading a centre and radius off an equation, and completing the square.", "meta": "6 questions · fluency + reasoning", "categories": [ "circles-coordinate-plane" ], "questions": [ { "id": "cg-013", "topic": "Equations of Circles", "difficulty": "easy", "stem": "Write the equation of the circle with centre (3, −5) and radius 4.", "choices": [ "(x − 3)² + (y − 5)² = 16", "(x + 3)² + (y − 5)² = 16", "(x − 3)² + (y + 5)² = 16", "(x − 3)² + (y + 5)² = 4" ], "correctIndex": 2, "explanations": [ "(y − 5) would put the centre at y = +5. The centre's y-coordinate is −5, and subtracting a negative gives (y + 5).", "The signs of both coordinates are flipped. The formula SUBTRACTS the centre coordinates, so a centre of (3, −5) gives (x − 3) and (y + 5).", "Correct. The form is (x − h)² + (y − k)² = r². With h = 3 and k = −5, y − (−5) becomes y + 5, and r² = 4² = 16.", "The right-hand side is the radius rather than the radius squared. The equation needs r², so 4 becomes 16." ], "hint": "In (x − h)² + (y − k)² = r², the centre is (h, k) — watch what happens to the sign when k is negative.", "solution": [ "Standard form is (x − h)² + (y − k)² = r².", "Here h = 3 and k = −5, and y − (−5) is written y + 5.", "r² = 4² = 16", "(x − 3)² + (y + 5)² = 16" ] }, { "id": "cg-014", "topic": "Equations of Circles", "difficulty": "easy", "stem": "A circle has the equation (x + 2)² + (y − 7)² = 49. What are its centre and radius?", "choices": [ "Centre (−2, 7), radius 7", "Centre (−2, 7), radius 49", "Centre (2, −7), radius 7", "Centre (2, −7), radius 49" ], "correctIndex": 0, "explanations": [ "Correct. (x + 2) gives h = −2, (y − 7) gives k = 7, and r = √49 = 7.", "The centre is right, but 49 is r², not r. Take the square root: r = 7.", "The signs are reversed. (x + 2) means x − (−2), so h = −2, not 2.", "Both the signs and the radius are off — read (x + 2) as x − (−2), and take the square root of 49." ], "solution": [ "(x + 2) is (x − (−2)), so h = −2.", "(y − 7) gives k = 7.", "r = √49 = 7", "Centre (−2, 7), radius 7" ] }, { "id": "cg-015", "topic": "Equations of Circles", "difficulty": "hard", "stem": "Find the centre and radius of the circle x² + y² − 6x + 8y − 11 = 0.", "choices": [ "Centre (3, −4), radius 6", "Centre (3, −4), radius 36", "Centre (−3, 4), radius 6", "Centre (6, −8), radius √11" ], "correctIndex": 0, "explanations": [ "Correct. Completing the square gives (x − 3)² + (y + 4)² = 11 + 9 + 16 = 36, so the centre is (3, −4) and r = 6.", "The centre is right, but 36 is r². The radius is √36 = 6.", "The signs are inverted. Completing the square on x² − 6x gives (x − 3)², which puts the centre at x = +3.", "(6, −8) reads the coefficients straight off the equation without completing the square. Each centre coordinate is HALF the coefficient, with the sign flipped." ], "hint": "Group the x terms and the y terms, then add the square of half each coefficient to both sides.", "solution": [ "Group the variables: (x² − 6x) + (y² + 8y) = 11.", "Complete both squares, adding 9 and 16 to each side.", "(x − 3)² + (y + 4)² = 11 + 9 + 16 = 36", "Centre (3, −4), radius 6" ] }, { "id": "cg-016", "topic": "Equations of Circles", "difficulty": "hard", "stem": "The endpoints of a diameter of a circle are (1, 2) and (7, 10). What is the equation of the circle?", "choices": [ "(x − 4)² + (y − 6)² = 25", "(x − 4)² + (y − 6)² = 100", "(x − 8)² + (y − 12)² = 25", "(x − 3)² + (y − 4)² = 25" ], "correctIndex": 0, "explanations": [ "Correct. The centre is the midpoint, (4, 6). The diameter has length √(6² + 8²) = 10, so the radius is 5 and r² = 25.", "100 uses the DIAMETER squared. The equation needs the radius squared, and the radius is half the diameter.", "(8, 12) is the SUM of the coordinates. The midpoint is the sum divided by two.", "(3, 4) is the difference between the endpoints, not their midpoint. The midpoint averages the coordinates." ], "hint": "The centre of a circle is the midpoint of any diameter, and the radius is half the diameter's length.", "solution": [ "The centre is the midpoint of the diameter: ((1 + 7)/2, (2 + 10)/2) = (4, 6).", "The diameter has length √(6² + 8²) = √100 = 10.", "The radius is half of that, 5, so r² = 25.", "(x − 4)² + (y − 6)² = 25" ] }, { "id": "cg-040", "topic": "Reasoning: Error Analysis", "difficulty": "hard", "stem": "A student converts x² + y² + 10x − 4y + 13 = 0 and writes the centre as (5, −2). What is the actual centre?", "choices": [ "(−10, 4)", "(−5, 2)", "(10, −4)", "(5, −2)" ], "correctIndex": 1, "explanations": [ "(−10, 4) uses the coefficients directly without halving them.", "Correct. x² + 10x becomes (x + 5)² and y² − 4y becomes (y − 2)², so the centre is (−5, 2). Each coordinate is half the coefficient with the sign reversed.", "(10, −4) neither halves nor reverses the signs.", "(5, −2) has both signs backwards. Completing the square on x² + 10x gives (x + 5)², which places the centre at x = −5." ], "hint": "Halve each coefficient, then flip its sign. Both steps, in that order.", "solution": [ "Complete the square in each variable: x² + 10x becomes (x + 5)², and y² − 4y becomes (y − 2)².", "Standard form reads (x − h)² + (y − k)², so (x + 5)² gives h = −5 and (y − 2)² gives k = 2.", "Each coordinate is half the coefficient with its sign reversed, which is the step the student skipped.", "Centre (−5, 2)" ], "kind": "reasoning" }, { "id": "cg-048", "topic": "Reasoning: Error Analysis", "difficulty": "medium", "stem": "A student writes the equation of the circle with centre (0, 4) and radius 6 as x² + (y − 4)² = 6. What is wrong?", "choices": [ "The right-hand side must be r², so it should be 36.", "The x term should be (x − 0)², which is not the same as x².", "Nothing is wrong.", "The centre's sign is wrong; it should be (y + 4)." ], "correctIndex": 0, "explanations": [ "Correct. The standard form is (x − h)² + (y − k)² = r². The radius is 6, so the right-hand side is 6² = 36, not 6.", "(x − 0)² is exactly x², so writing it either way is fine.", "Leaving 6 rather than 36 describes a circle of radius √6, which is a different circle.", "(y − 4) is correct for a centre with k = 4 — the formula subtracts the centre coordinate." ], "hint": "Which side of the equation holds the radius, and in what form?", "solution": [ "Standard form is (x − h)² + (y − k)² = r².", "The right-hand side holds r², not r.", "r = 6, so r² = 36.", "x² + (y − 4)² = 36" ], "kind": "reasoning" } ] }, { "id": "circles-inscribed-angles", "title": "Inscribed Angles & Cyclic Quadrilaterals", "blurb": "Half the arc, Thales' theorem, and the supplementary opposite angles of a cyclic quadrilateral.", "meta": "9 questions · fluency + reasoning", "categories": [ "circles" ], "questions": [ { "id": "cg-017", "topic": "Inscribed Angles", "difficulty": "easy", "stem": "Inscribed angle ∠ABC measures 35° in circle O. What is the measure of its intercepted arc AC?", "choices": [ "17.5°", "35°", "70°", "145°" ], "correctIndex": 2, "explanations": [ "17.5° halves the angle. The halving goes the other way: the ANGLE is half the arc, so the arc is twice the angle.", "35° treats this like a central angle. The vertex here is on the circle, not at the centre, so the angle and arc are not equal.", "Correct. An inscribed angle is half its intercepted arc, so the arc is twice the angle: 2 × 35° = 70°.", "145° is 180° − 35°. Supplements come up with cyclic quadrilaterals, not with a single inscribed angle." ], "hint": "The vertex sits on the circle. Which is bigger — the inscribed angle, or the arc it intercepts?", "solution": [ "An inscribed angle is half its intercepted arc, so the arc is twice the angle.", "arc AC = 2 × 35°", "arc AC = 70°" ], "diagram": "BAC35°?", "diagramCaption": "Not drawn to scale" }, { "id": "cg-018", "topic": "Inscribed Angles", "difficulty": "easy", "stem": "In circle O, arc RS measures 140°. Point T lies on the major arc. What is the measure of inscribed angle ∠RTS?", "choices": [ "35°", "70°", "140°", "220°" ], "correctIndex": 1, "explanations": [ "35° divides by four. The inscribed angle is half the arc, which requires one division by two, not two.", "Correct. An inscribed angle is half its intercepted arc: 140° ÷ 2 = 70°.", "140° is the arc itself. Only a CENTRAL angle equals its arc; an inscribed angle is half of it.", "220° is the major arc (360° − 140°). The angle intercepts arc RS, and in any case an inscribed angle is smaller than its arc, never larger." ], "solution": [ "An inscribed angle is half its intercepted arc.", "∠RTS = 140° ÷ 2", "∠RTS = 70°" ] }, { "id": "cg-019", "topic": "Inscribed Angles", "difficulty": "easy", "stem": "In circle O, segment AB is a diameter and C is a point on the circle, distinct from A and B. What is the measure of ∠ACB?", "choices": [ "It depends on where C is", "90°", "45°", "60°" ], "correctIndex": 1, "explanations": [ "It genuinely does not depend on C. Wherever C sits on the circle, the intercepted arc is the same semicircle, so the angle is always 90° — which is exactly what makes this theorem useful.", "Correct. AB is a diameter, so it intercepts a semicircle of 180°. The inscribed angle is half of that: 90°. This is Thales' theorem — any angle inscribed in a semicircle is right.", "45° would be correct only if the intercepted arc were 90°. A diameter cuts off a semicircle of 180°.", "60° would need an intercepted arc of 120°, which a diameter does not produce." ], "hint": "What arc does a diameter intercept, and what is half of it?", "solution": [ "AB is a diameter, so it cuts off a semicircle: an arc of 180°.", "∠ACB is inscribed in that semicircle, so it is half the arc.", "∠ACB = 180° ÷ 2 = 90°", "This is Thales' theorem: any angle inscribed in a semicircle is right." ], "diagram": "ABCO?" }, { "id": "cg-020", "topic": "Inscribed Angles", "difficulty": "medium", "stem": "Quadrilateral ABCD is inscribed in a circle. If ∠A = 105°, what is the measure of ∠C?", "choices": [ "52.5°", "75°", "105°", "255°" ], "correctIndex": 1, "explanations": [ "52.5° halves the given angle. Halving relates an inscribed angle to its arc, not one opposite angle to another.", "Correct. Opposite angles of a cyclic quadrilateral are supplementary, so ∠C = 180° − 105° = 75°.", "105° would make the opposite angles equal. That is true of a parallelogram, but not of a quadrilateral inscribed in a circle.", "255° is 360° − 105°. The four angles of a quadrilateral total 360°, but OPPOSITE pairs total 180° each." ], "hint": "∠A and ∠C intercept arcs that together make the whole circle. What does that make the two angles sum to?", "solution": [ "Opposite angles of a cyclic quadrilateral are supplementary.", "∠C = 180° − 105°", "∠C = 75°" ] }, { "id": "cg-036", "topic": "Reasoning: Error Analysis", "difficulty": "hard", "stem": "An inscribed angle in a circle measures 50°. A student writes: \"The intercepted arc is also 50°, because the angle and the arc are the same.\" What went wrong, and what is the arc?", "choices": [ "Nothing went wrong; the arc is 50°.", "They used the central-angle relationship on an inscribed angle. The arc is 100°.", "They forgot the arc is a major arc. The arc is 310°.", "They should have used supplements. The arc is 130°." ], "correctIndex": 1, "explanations": [ "The arc is not 50°. \"Angle equals arc\" holds only when the vertex is at the CENTRE.", "Correct. Angle = arc is the central-angle relationship, and it needs the vertex at the centre. An inscribed angle has its vertex on the circle and measures half its arc, so the arc is 2 × 50° = 100°.", "Major and minor is not the issue — the student's error was applying the wrong relationship, and the intercepted arc here is the minor arc of 100°.", "Supplements relate opposite angles of a cyclic quadrilateral. There is no quadrilateral here, and the relationship is a doubling." ], "hint": "Ask first where the vertex is. That single fact decides which relationship applies.", "solution": [ "Angle = arc is the CENTRAL-angle relationship, and it needs the vertex at the centre.", "This angle is inscribed: its vertex is on the circle, so it measures half its arc.", "arc = 2 × 50° = 100°" ], "kind": "reasoning" }, { "id": "cg-039", "topic": "Reasoning: Choose the Theorem", "difficulty": "medium", "stem": "In circle O, points A and B lie on the circle and C and D are two other points on the major arc AB. Which reason explains why ∠ACB ≅ ∠ADB?", "choices": [ "They are vertical angles.", "Both are inscribed angles intercepting the same arc AB, so each is half of it.", "Both are central angles.", "ACBD is a parallelogram, so opposite angles are congruent." ], "correctIndex": 1, "explanations": [ "Vertical angles are formed by two intersecting lines. These angles have different vertices.", "Correct. Each angle has its vertex on the circle and intercepts arc AB, so each measures half of arc AB — which makes them equal regardless of where C and D sit.", "A central angle has its vertex at O. Both of these have vertices on the circle.", "ACBD need not be a parallelogram, and opposite angles of a CYCLIC quadrilateral are supplementary rather than congruent." ], "hint": "Both angles are half the same thing.", "solution": [ "Both angles have their vertex on the circle, so both are inscribed angles.", "Both intercept the same arc AB.", "Each is therefore half of arc AB, which makes them equal wherever C and D sit." ], "kind": "reasoning" }, { "id": "cg-042", "topic": "Reasoning: Error Analysis", "difficulty": "medium", "stem": "Arc PQ measures 96°. A student finds the inscribed angle by computing 96° × 2 = 192°. Without doing any arithmetic, how can you tell this is wrong?", "choices": [ "The arc should have been 360° − 96° first.", "Inscribed angles are always right angles.", "An inscribed angle can never be larger than 180°, and it must be smaller than its arc.", "The student should have added 96° and 180°." ], "correctIndex": 2, "explanations": [ "The minor arc is the intercepted one here, and switching to the major arc would not fix a doubling that should have been a halving.", "Only angles inscribed in a semicircle are right. That is a special case, not the general rule.", "Correct. An inscribed angle is HALF its arc, so it is always smaller than the arc — and an angle of 192° is impossible in a triangle-like figure regardless. The answer is 48°.", "Adding is not the relationship. The inscribed angle is half the arc." ], "hint": "You do not need the number — just ask whether the answer is even possible.", "solution": [ "An inscribed angle is HALF its arc, so it is always smaller than the arc it intercepts.", "Multiplying by 2 makes it larger than the arc, which is impossible — no arithmetic needed to see it.", "The angle is 96° ÷ 2 = 48°." ], "kind": "reasoning" }, { "id": "cg-043", "topic": "Reasoning: Error Analysis", "difficulty": "medium", "stem": "Quadrilateral WXYZ is inscribed in a circle. A student claims ∠W and ∠X must be supplementary because \"opposite angles of a cyclic quadrilateral add to 180°.\" What is wrong?", "choices": [ "The theorem only applies to squares inscribed in circles.", "Nothing is wrong.", "The theorem is wrong; cyclic quadrilateral angles are congruent, not supplementary.", "The theorem is right, but ∠W and ∠X are ADJACENT. The supplementary pairs are ∠W with ∠Y, and ∠X with ∠Z." ], "correctIndex": 3, "explanations": [ "It applies to every quadrilateral inscribed in a circle, not only to squares.", "The theorem was quoted correctly but applied to the wrong pair of angles.", "The theorem is correctly stated — opposite angles of a cyclic quadrilateral are indeed supplementary.", "Correct. In WXYZ the vertices are in order, so W and X are next to each other. Opposite pairs are W–Y and X–Z, and those are the ones that sum to 180°." ], "hint": "Read the vertex order in the name of the quadrilateral. Which vertices are actually opposite?", "solution": [ "The theorem quoted is correct: opposite angles of a cyclic quadrilateral sum to 180°.", "In WXYZ the vertices are named in order, so W and X are ADJACENT, not opposite.", "The supplementary pairs are ∠W with ∠Y, and ∠X with ∠Z." ], "kind": "reasoning" }, { "id": "cg-045", "topic": "Reasoning: Choose the Theorem", "difficulty": "medium", "stem": "A triangle is inscribed in a circle so that one of its sides is a diameter. What can be concluded about the triangle, and on what grounds?", "choices": [ "Nothing can be concluded without knowing the radius.", "It is a right triangle, because an angle inscribed in a semicircle intercepts a 180° arc and is therefore 90°.", "It is equilateral, because all radii are congruent.", "It is isosceles, because two sides are radii." ], "correctIndex": 1, "explanations": [ "The conclusion holds for every circle, whatever the radius — which is precisely what makes the theorem useful.", "Correct. This is Thales' theorem. The angle opposite the diameter is inscribed in a semicircle, intercepting an arc of 180°, so it measures half of that: 90°.", "Radii being congruent does not make the triangle's SIDES congruent — two of the triangle's sides are chords, not radii.", "The two sides meeting at the third vertex are chords, not radii, and they are generally different lengths." ], "hint": "What arc does the diameter cut off, and what is half of it?", "solution": [ "The side that is a diameter cuts off a semicircle — an arc of 180°.", "The angle opposite that side is inscribed in the semicircle, so it is half the arc.", "That angle is 90°, so the triangle is right-angled. This is Thales' theorem." ], "kind": "reasoning" } ] }, { "id": "circles-secant-tangent-angles", "title": "Secant & Tangent Angles", "blurb": "Vertex inside, on, or outside the circle — and which of add, halve or subtract each one calls for.", "meta": "10 questions · fluency + reasoning", "categories": [ "circles" ], "questions": [ { "id": "cg-021", "topic": "Chord & Secant Angles", "difficulty": "medium", "stem": "Two chords intersect inside a circle. The two arcs they intercept measure 80° and 40°. What is the measure of the angle formed at the intersection?", "choices": [ "20°", "40°", "60°", "120°" ], "correctIndex": 2, "explanations": [ "20° is half the DIFFERENCE of the arcs. Halving a difference is the rule for a vertex OUTSIDE the circle; this vertex is inside.", "40° is one of the given arcs, not a combination of them.", "Correct. An angle formed by two chords meeting inside a circle is half the SUM of the two intercepted arcs: ½(80° + 40°) = 60°.", "120° is the sum of the arcs without halving. The formula is half the sum." ], "hint": "Vertex inside the circle: add the two arcs. Vertex outside: subtract them. Either way, halve the result.", "solution": [ "An angle formed where two chords meet INSIDE a circle is half the SUM of the two intercepted arcs.", "angle = ½(80° + 40°) = ½(120°)", "angle = 60°" ], "diagram": "ABCD80°40°?" }, { "id": "cg-022", "topic": "Chord & Secant Angles", "difficulty": "medium", "stem": "Two secants are drawn to a circle from an external point P. They intercept a far arc of 110° and a near arc of 40°. What is the measure of ∠P?", "choices": [ "35°", "55°", "70°", "75°" ], "correctIndex": 0, "explanations": [ "Correct. An angle formed outside the circle is half the DIFFERENCE of the intercepted arcs: ½(110° − 40°) = 35°.", "55° is half of the far arc alone, ignoring the near arc entirely.", "70° is the difference of the arcs without halving.", "75° is half the SUM of the arcs. Adding is the rule for a vertex inside the circle; an external vertex takes the difference." ], "hint": "The vertex is outside the circle. Does that call for the sum of the arcs or the difference?", "solution": [ "A vertex OUTSIDE the circle takes half the DIFFERENCE of the intercepted arcs.", "∠P = ½(110° − 40°) = ½(70°)", "∠P = 35°" ], "diagram": "P110°40°?", "diagramCaption": "Not drawn to scale" }, { "id": "cg-023", "topic": "Chord & Secant Angles", "difficulty": "hard", "stem": "Two secants meet at an external point, forming a 25° angle. The nearer intercepted arc measures 50°. What is the measure of the farther arc?", "choices": [ "25°", "75°", "100°", "125°" ], "correctIndex": 2, "explanations": [ "25° is the angle itself, not an arc.", "75° comes from adding the angle to the near arc. The relationship involves half the difference, so the angle must be doubled before comparing arcs.", "Correct. 25° = ½(far − 50°), so far − 50° = 50°, giving far = 100°.", "125° adds the doubled angle to the near arc but then adds the angle once more. Double 25° to get 50°, then add the near arc: 50° + 50° = 100°." ], "hint": "Write the relationship as an equation and solve for the unknown arc rather than trying to do it in one step.", "solution": [ "Outside the circle, angle = ½(far arc − near arc).", "25° = ½(far − 50°)", "far − 50° = 50°", "far arc = 100°" ] }, { "id": "cg-024", "topic": "Chord & Secant Angles", "difficulty": "medium", "stem": "Two chords intersect inside a circle, forming a 70° angle. One of the intercepted arcs measures 90°. What is the measure of the other intercepted arc?", "choices": [ "20°", "50°", "80°", "140°" ], "correctIndex": 1, "explanations": [ "20° comes from 90° − 70°, subtracting the angle directly from the arc. The angle is half the SUM of the arcs, so it must be doubled first.", "Correct. 70° = ½(90° + x), so 140° = 90° + x, giving x = 50°.", "80° does not satisfy the relationship: ½(90° + 80°) = 85°, not 70°.", "140° is twice the angle, which equals the SUM of both arcs — the second arc is what remains after subtracting the first." ], "hint": "Double the angle first. That gives you the sum of the two arcs.", "solution": [ "Inside the circle, angle = ½(one arc + the other).", "70° = ½(90° + x)", "140° = 90° + x", "x = 50°" ] }, { "id": "cg-025", "topic": "Tangent Angles", "difficulty": "medium", "stem": "A tangent and a chord meet at a point on a circle. The chord cuts off an arc of 130° on the side of the angle. What is the measure of the angle between the tangent and the chord?", "choices": [ "32.5°", "65°", "115°", "130°" ], "correctIndex": 1, "explanations": [ "32.5° divides by four. The tangent–chord angle is half the arc, which is a single division by two.", "Correct. A tangent–chord angle equals half its intercepted arc: ½(130°) = 65°.", "115° is 180° − 65°, the angle on the OTHER side of the tangent line. That angle intercepts the other arc.", "130° is the arc itself. Only a central angle equals its arc." ], "hint": "A tangent–chord angle follows the same half-the-arc rule as an inscribed angle — think of the tangent as a chord whose second endpoint has slid into T.", "solution": [ "A tangent–chord angle equals half its intercepted arc.", "angle = ½(130°)", "angle = 65°" ], "diagram": "TC?130°", "diagramCaption": "The horizontal line is tangent at T" }, { "id": "cg-026", "topic": "Tangent Angles", "difficulty": "medium", "stem": "From an external point, a secant and a tangent are drawn to a circle. They intercept a far arc of 150° and a near arc of 60°. What is the measure of the angle at the external point?", "choices": [ "45°", "75°", "90°", "105°" ], "correctIndex": 0, "explanations": [ "Correct. Any angle with its vertex outside the circle is half the difference of the intercepted arcs: ½(150° − 60°) = 45°.", "75° is half the far arc alone. Both arcs matter.", "90° is the difference of the arcs without halving.", "105° is half the SUM. Summing applies to a vertex inside the circle; this vertex is outside." ], "solution": [ "Any angle with its vertex outside the circle is half the difference of the intercepted arcs.", "angle = ½(150° − 60°) = ½(90°)", "angle = 45°" ] }, { "id": "cg-027", "topic": "Tangent Angles", "difficulty": "medium", "stem": "Two tangents are drawn to a circle from the same external point. They intercept a major arc of 210° and a minor arc of 150°. What is the measure of the angle between the tangents?", "choices": [ "30°", "60°", "90°", "180°" ], "correctIndex": 0, "explanations": [ "Correct. The vertex is outside the circle, so the angle is half the difference: ½(210° − 150°) = 30°.", "60° is the difference of the arcs without halving.", "90° would require the arcs to differ by 180°. They differ by 60°.", "180° is the sum of the arcs halved (½ × 360°), which would mean the two tangents form a straight line — impossible for two tangents meeting at a point." ], "hint": "Two tangents from one point still make an external angle, so the same half-the-difference rule applies.", "solution": [ "The vertex is outside the circle, so take half the difference of the two arcs.", "angle = ½(210° − 150°) = ½(60°)", "angle = 30°" ] }, { "id": "cg-028", "topic": "Tangent Angles", "difficulty": "hard", "stem": "Two tangents from an external point form a 40° angle. What is the measure of the minor arc between the two points of tangency?", "choices": [ "40°", "80°", "140°", "220°" ], "correctIndex": 2, "explanations": [ "40° is the angle itself. The angle and the minor arc are not equal for an external vertex.", "80° is twice the angle, which gives the DIFFERENCE between the two arcs — not the minor arc itself.", "Correct. The arcs sum to 360° and differ by 2 × 40° = 80°. Solving gives major = 220° and minor = 140°.", "220° is the MAJOR arc. The question asks for the minor one." ], "hint": "Set up two facts about the arcs: they add to 360°, and their difference is twice the angle.", "solution": [ "The two arcs make a full circle: major + minor = 360°.", "The vertex is outside, so 40° = ½(major − minor), giving major − minor = 80°.", "Adding the two equations: 2 × major = 440°, so major = 220°.", "minor arc = 360° − 220° = 140°" ] }, { "id": "cg-037", "topic": "Reasoning: Error Analysis", "difficulty": "hard", "stem": "Two secants meet outside a circle, intercepting arcs of 120° and 40°. A student computes the angle as ½(120° + 40°) = 80°. What is the error?", "choices": [ "The arcs should be added but not halved; the angle is 160°.", "An external vertex takes half the DIFFERENCE, not half the sum; the angle is 40°.", "The student should have used only the far arc; the angle is 60°.", "There is no error; the angle is 80°." ], "correctIndex": 1, "explanations": [ "Halving is correct — it is the addition that is wrong. External angles use the difference.", "Correct. Adding the arcs is the rule for a vertex INSIDE the circle. Outside the circle the angle is half the difference: ½(120° − 40°) = 40°.", "Both arcs are needed. Using only the far arc would give 60°, which ignores how far outside the circle the vertex sits.", "80° is what the inside-the-circle rule gives. The vertex here is outside, so the rule does not apply." ], "hint": "Inside the circle: add. Outside the circle: subtract. Which is this?", "solution": [ "Adding the two arcs is the rule for a vertex INSIDE the circle.", "This vertex is outside, so the angle is half their difference.", "angle = ½(120° − 40°) = 40°" ], "kind": "reasoning" }, { "id": "cg-050", "topic": "Reasoning: Choose the Theorem", "difficulty": "hard", "stem": "In circle O, a tangent meets a chord at point T, forming a 40° angle. Separately, an inscribed angle elsewhere on the circle intercepts the same arc. What is that inscribed angle, and why?", "choices": [ "20°, because an inscribed angle is half a tangent–chord angle.", "50°, because they are complementary.", "40°, because both a tangent–chord angle and an inscribed angle equal half their intercepted arc.", "80°, because the inscribed angle doubles the tangent–chord angle." ], "correctIndex": 2, "explanations": [ "There is no halving between the two — both are already half of the same arc, so they are equal.", "Complementary angles sum to 90°, which has no bearing on these two.", "Correct. A tangent–chord angle is half its intercepted arc, and so is an inscribed angle intercepting that same arc. Both equal ½(80°) = 40°.", "Doubling the tangent–chord angle gives the ARC (80°), not the inscribed angle." ], "hint": "Work out the arc from the tangent–chord angle first, then use it for the inscribed angle.", "solution": [ "A tangent–chord angle is half its intercepted arc, so that arc is 2 × 40° = 80°.", "An inscribed angle intercepting the same arc is also half of it.", "inscribed angle = ½(80°) = 40°" ], "kind": "reasoning" } ], "studyGuide": { "mentalModel": { "corePrinciple": "Every angle whose sides meet a circle is half of a signed combination of the arcs it looks at: θ = ½(far arc ± near arc). Where the vertex sits decides the sign and the count. At the centre: one arc, whole. On the circle: one arc, halved. Inside: two arcs, added, halved. Outside: two arcs, subtracted, halved. There is one theorem here, seen from four positions.", "firstPrinciplesDerivation": "Nothing is assumed beyond two facts about triangles: a triangle with two radii for sides is isosceles, and an exterior angle of a triangle equals the sum of the two remote interior angles. Put an inscribed angle's vertex P on the circle and draw the diameter through P. Each side closes an isosceles triangle with the centre O, and the exterior angle at O is twice the base angle at P; adding the two halves gives ∠APB = ½ arc AB. That one statement is the engine. Every other case is produced by drawing one chord to make a triangle whose exterior angle is the angle you want, and whose remote interior angles are two inscribed angles you already know how to measure. The ± is nothing more than whether the exterior-angle theorem is read forwards (vertex inside: the angle IS the exterior angle, so it is a sum) or backwards (vertex outside: the angle is a remote interior angle, so it is a difference)." }, "conceptMatrix": [ { "caseName": "Central angle — vertex at the centre", "spatialConfiguration": "Vertex at O; both sides are radii ending on the circle at A and B.", "governingFormula": "∠AOB = arc AB", "auxiliaryLineProof": "No auxiliary line — this is the definition that grounds everything else. The measure of an arc is defined to be the measure of the central angle that subtends it, and every later formula is proved by reducing an angle to a combination of central angles.", "svgIllustration": "OABθ = arc" }, { "caseName": "Inscribed angle and tangent–chord angle — vertex on the circle", "spatialConfiguration": "Vertex P on the circle. Inscribed: both sides are chords PA and PB, and the angle intercepts the arc AB that does NOT contain P. Tangent–chord: one side is a chord, the other is the tangent at P; the arc is the one inside the angle. Same position, same rule.", "governingFormula": "∠APB = ½ arc AB (and the tangent–chord angle at T = ½ arc TC)", "auxiliaryLineProof": "Draw the radii OA and OB and the diameter through P. Triangle OPA is isosceles (OP = OA), so its base angles are equal, call each x; the exterior angle at O is 2x. Triangle OPB likewise gives 2y. So ∠AOB = 2x + 2y = 2∠APB, and since ∠AOB = arc AB, ∠APB = ½ arc AB. For the tangent–chord angle draw the radius to the point of tangency T: it is perpendicular to the tangent, and triangle OTC is isosceles with apex angle arc TC, so its base angle at T is 90° − ½ arc TC, and the angle between tangent and chord is the complement, ½ arc TC. It is the inscribed case in the limit where the second chord's far end slides into T — the formula does not change, which is why the two share a row.", "svgIllustration": "POABtangent at Pθ = ½ arc" }, { "caseName": "Two chords crossing — vertex inside the circle", "spatialConfiguration": "Vertex E strictly inside the circle, where chords AB and CD cross. The angle ∠AEC and its vertical angle ∠BED intercept the two arcs AC and BD lying inside them.", "governingFormula": "∠AEC = ½(arc AC + arc BD)", "auxiliaryLineProof": "Draw the chord AD, forming triangle AED. ∠AEC is an exterior angle of that triangle at E, so it equals the sum of the two remote interior angles, ∠EAD + ∠EDA. Both are inscribed angles: ∠EAD = ½ arc BD and ∠EDA = ½ arc AC. Adding gives the formula. The inside case is the exterior-angle theorem read forwards, applied to two inscribed angles.", "svgIllustration": "EACDBθ = ½(arc + arc)" }, { "caseName": "Secants and tangents from an external point — vertex outside the circle", "spatialConfiguration": "Vertex P outside the circle. Each side cuts the circle at a near point and a far point (a tangent side touches at one point that plays both roles). The angle intercepts a far arc and a near arc. Two tangents are the extreme case: near and far together make the whole circle.", "governingFormula": "∠P = ½(far arc − near arc) (two tangents: ∠P = 180° − near arc)", "auxiliaryLineProof": "Let the secants be PAB and PCD with A, C near and B, D far. Draw the chord BC, forming triangle PBC. ∠PBC is the inscribed angle on the near arc, ½ arc AC; the exterior angle of the triangle at C, ∠BCD, is the inscribed angle on the far arc, ½ arc BD. The exterior-angle theorem says ∠BCD = ∠P + ∠PBC, so ∠P = ½ arc BD − ½ arc AC. This is the same theorem as the inside case, read backwards: the target angle is now a remote interior angle, so it is a difference. For two tangents the arcs sum to 360°, so ∠P = ½(360° − 2·near) = 180° − near arc.", "svgIllustration": "PBDACfarnearθ = ½(far − near)" } ], "diagnosticPitfalls": [ { "misconception": "Inside and outside the circle use the same formula, so the arcs are added for an external vertex (question 9: ½(120° + 40°) = 80° for two secants from an outside point).", "rootCause": "The two formulas look alike — ½ of two arcs — and get memorised as one template with an arbitrary sign, instead of as the exterior-angle theorem read in two directions.", "diagnosticCue": "Slide the vertex outward along a chord until it leaves the circle. Does the angle grow or shrink? It shrinks, and keeps shrinking as the vertex recedes. A formula that adds arcs can never give less than half the far arc, so it cannot describe an angle heading toward 0°: the outside case must subtract. Here: ½(120° − 40°) = 40°." }, { "misconception": "An inscribed angle is half of a tangent–chord angle on the same arc, so a 40° tangent–chord angle means a 20° inscribed angle (question 10).", "rootCause": "Over-halving: the rule \"half the arc\" gets applied a second time, to an angle that is already half the arc, because the tangent–chord angle is not recognised as the same kind of object as the inscribed angle.", "diagnosticCue": "Ask what each angle is half OF. Both are half of the same arc, so they are equal — 40° and 40°, with the arc itself 80°. Halving only ever goes from an ARC to an angle, never from one angle to another. If the answer you are writing is half of an angle, you have halved twice." }, { "misconception": "The angle equals the arc it intercepts, wherever the vertex is.", "rootCause": "Surface-level pattern matching from the central angle, the first case anyone meets: the only prior experience of \"angle and arc\" was the one where they are equal, and the vertex moving off the centre is not felt as changing anything.", "diagnosticCue": "Put the vertex on the circle with A and B the ends of a diameter. The arc is 180° and the angle is visibly a right angle, so the angle cannot equal the arc. Once the semicircle is seen, the halving is felt rather than memorised." } ], "workedExamples": [ { "title": "Question 4 — two chords inside, one arc unknown", "problemStatement": "Two chords intersect inside a circle, forming a 70° angle. One of the intercepted arcs measures 90°. What is the measure of the other intercepted arc?", "causalAnalysis": "The vertex is strictly inside the circle, so the angle is the exterior angle of the triangle made by drawing the chord that joins the two arcs' endpoints — a SUM, then halved. Because the angle is known and an arc is missing, the first algebraic move is to undo the half: double the angle, and you are holding the sum of the two arcs.", "algebraicDerivation": [ "Step 1: Write the inside-vertex relationship with the unknown arc x: 70° = ½(90° + x).", "Step 2: Clear the half — double both sides: 140° = 90° + x. The 140° is the total of the two intercepted arcs.", "Step 3: Subtract the known arc: x = 140° − 90° = 50°. Check: ½(90° + 50°) = 70°. The two remaining arcs must total 360° − 140° = 220°, half of which is the supplementary angle 110°, consistent with 70° + 110° = 180°." ], "keyTakeaway": "Inside the circle, double the angle first: 2θ is the sum of the two arcs, and the rest is a subtraction. The distractor 20° (= 90° − 70°) is what subtracting BEFORE doubling produces — the angle is half a sum, so it is never one of the addends." }, { "title": "Question 8 — two tangents, minor arc from the angle alone", "problemStatement": "Two tangents from an external point form a 40° angle. What is the measure of the minor arc between the two points of tangency?", "causalAnalysis": "The vertex is outside, so the angle is half the DIFFERENCE of the far and near arcs. With two tangents there are only two arcs and together they make the whole circle, which supplies the second equation the unknowns need: their sum is 360°. Two facts about two arcs — a difference from the angle, a sum from the circle — is a two-equation system, and recognising that is the whole move.", "algebraicDerivation": [ "Step 1: Difference from the angle: 40° = ½(major − minor), so major − minor = 80°.", "Step 2: Sum from the circle: major + minor = 360°.", "Step 3: Add the equations: 2·major = 440°, so major = 220° and minor = 360° − 220° = 140°. Check: ½(220° − 140°) = 40°. Shortcut, same result: for two tangents the angle and the minor arc are supplementary, 180° − 40° = 140°." ], "keyTakeaway": "When only one arc measure is given (or none), look for the second equation the figure hands you for free — here, that two arcs fill the circle. The distractor 80° is the difference of the arcs, a correct intermediate mistaken for the answer; 220° is the major arc, the right system solved for the wrong unknown." } ] } }, { "id": "circles-segments-tangents", "title": "Segment Lengths & Tangents", "blurb": "Power of a point for chords, secants and tangents, and the right angle a tangent makes with a radius.", "meta": "9 questions · fluency + reasoning", "categories": [ "circles" ], "questions": [ { "id": "cg-029", "topic": "Segment Lengths", "difficulty": "medium", "stem": "Two chords intersect inside a circle. One is divided into segments of length 6 and 8; the other into segments of length 4 and x. What is x?", "choices": [ "2", "10", "12", "48" ], "correctIndex": 2, "explanations": [ "2 comes from dividing 8 by 4. The products of the two pieces of each chord are equal, so set 6 × 8 equal to 4 × x.", "10 comes from 6 + 8 − 4, combining the lengths by addition. The relationship is multiplicative.", "Correct. The two chords satisfy 6 × 8 = 4 × x, so 48 = 4x and x = 12.", "48 is the product 6 × 8, which equals 4x — not x itself. Divide by 4 to finish." ], "hint": "For two chords crossing inside a circle, the product of one chord's two pieces equals the product of the other's.", "solution": [ "Where two chords cross inside a circle, the products of the two pieces are equal.", "6 × 8 = 4 × x", "48 = 4x", "x = 12" ], "diagram": "684x" }, { "id": "cg-030", "topic": "Segment Lengths", "difficulty": "hard", "stem": "From external point P, one secant has external segment 4 and total length 15. A second secant from P has external segment 5. What is the total length of the second secant?", "choices": [ "7", "12", "16", "18.75" ], "correctIndex": 1, "explanations": [ "7 is the far portion of the second secant (12 − 5), not its total length. The relationship uses the WHOLE secant.", "Correct. For two secants from the same point, (external)(whole) is equal for both: 4 × 15 = 5 × whole, so whole = 60 ÷ 5 = 12.", "16 does not satisfy 5 × whole = 60.", "18.75 divides 75 by 4, mixing up which numbers pair together. The known secant contributes 4 × 15 = 60." ], "hint": "Each secant contributes (external part) × (whole secant), and the two products are equal.", "solution": [ "For two secants from the same external point, (external)(whole) is the same for both.", "4 × 15 = 5 × whole", "60 = 5 × whole", "whole = 12" ] }, { "id": "cg-031", "topic": "Segment Lengths", "difficulty": "hard", "stem": "A tangent of length 12 and a secant are drawn to a circle from the same external point. The secant's external segment measures 8. What is the length of the secant's internal chord?", "choices": [ "1.5", "4", "10", "18" ], "correctIndex": 2, "explanations": [ "1.5 divides 12 by 8 without squaring the tangent. The tangent length appears squared in this relationship.", "4 is 12 − 8, subtracting the lengths directly. The relationship is multiplicative, not additive.", "Correct. The tangent-secant relationship gives 12² = 8 × (whole), so the whole secant is 144 ÷ 8 = 18. The internal chord is 18 − 8 = 10.", "18 is the WHOLE secant. The internal chord is what remains after removing the external segment of 8." ], "hint": "The tangent squared equals the external segment times the WHOLE secant — then read carefully which piece is asked for.", "solution": [ "Tangent and secant from one point: (tangent)² = (external)(whole secant).", "12² = 8 × whole, so whole = 144 ÷ 8 = 18.", "The internal chord is the whole secant minus its external part.", "chord = 18 − 8 = 10" ] }, { "id": "cg-032", "topic": "Segment Lengths", "difficulty": "medium", "stem": "Two chords intersect inside a circle. One is split into segments of 9 and x; the other into segments of 6 and 12. What is x?", "choices": [ "3", "8", "15", "72" ], "correctIndex": 1, "explanations": [ "3 comes from 9 − 6, subtracting the pieces. The relationship multiplies them.", "Correct. 9x = 6 × 12 = 72, so x = 8.", "15 comes from adding 6 and 12 and subtracting 3. The two chords relate by equal products, not by sums.", "72 is the product 6 × 12, which equals 9x — divide by 9 to get x." ], "solution": [ "The products of the two pieces of each chord are equal.", "9x = 6 × 12 = 72", "x = 8" ] }, { "id": "cg-033", "topic": "Tangents", "difficulty": "medium", "stem": "PA is tangent to circle O at point A. The radius OA measures 9 and the tangent PA measures 12. What is the distance OP?", "choices": [ "15", "3", "√63", "21" ], "correctIndex": 0, "explanations": [ "Correct. A tangent is perpendicular to the radius at the point of tangency, so OAP is a right triangle with legs 9 and 12. OP = √(81 + 144) = √225 = 15.", "3 is 12 − 9. The three lengths form a right triangle, not a straight line.", "√63 comes from 144 − 81, subtracting the squares. That would find a LEG; here the unknown is the hypotenuse, so the squares are added.", "21 adds the two lengths. They are the legs of a right triangle, so they combine by the Pythagorean theorem instead." ], "hint": "What is the angle between a tangent and the radius drawn to its point of tangency?", "solution": [ "A tangent is perpendicular to the radius at the point of tangency.", "So triangle OAP is right-angled at A, with legs 9 and 12.", "OP = √(9² + 12²) = √(81 + 144) = √225", "OP = 15" ], "diagram": "OAP912?", "diagramCaption": "A tangent meets the radius at its point of tangency" }, { "id": "cg-034", "topic": "Tangents", "difficulty": "medium", "stem": "Two tangent segments are drawn to a circle from external point Q. One measures 3x − 2 and the other measures x + 8. What is the value of x?", "choices": [ "3", "5", "6", "10" ], "correctIndex": 1, "explanations": [ "3 gives tangent lengths of 7 and 11, which are not equal.", "Correct. Two tangent segments from the same external point are congruent, so 3x − 2 = x + 8. Then 2x = 10 and x = 5, making both tangents 13.", "6 gives lengths of 16 and 14 — close, but not equal.", "10 gives 28 and 18, which are not equal. Set the two expressions equal to each other and solve." ], "hint": "What is always true about two tangent segments drawn to a circle from the same outside point?", "solution": [ "Two tangent segments drawn from the same external point are congruent.", "3x − 2 = x + 8", "2x = 10", "x = 5" ] }, { "id": "cg-035", "topic": "Tangents", "difficulty": "medium", "stem": "PT is tangent to circle O at T. If the radius of the circle is 8 and OP = 17, what is the length of the tangent segment PT?", "choices": [ "9", "15", "25", "√353" ], "correctIndex": 1, "explanations": [ "9 is 17 − 8, subtracting the lengths directly. They are sides of a right triangle, not points on a line.", "Correct. The tangent is perpendicular to the radius, so PT = √(17² − 8²) = √(289 − 64) = √225 = 15.", "25 adds the two given lengths. The Pythagorean relationship applies instead.", "√353 comes from ADDING the squares. Here the hypotenuse (OP = 17) is known and a leg is unknown, so the squares are subtracted." ], "solution": [ "The tangent is perpendicular to the radius, so OTP is right-angled at T.", "PT = √(17² − 8²) = √(289 − 64) = √225", "PT = 15" ] }, { "id": "cg-038", "topic": "Reasoning: Choose the Theorem", "difficulty": "medium", "stem": "Two tangent segments are drawn to a circle from the same external point. Which statement justifies concluding that the two segments are congruent?", "choices": [ "Chords equidistant from the centre are congruent.", "The two right triangles formed with the radii are congruent by Hypotenuse-Leg.", "Inscribed angles that intercept the same arc are congruent.", "The two tangents intercept congruent arcs." ], "correctIndex": 1, "explanations": [ "This is a real theorem, but it is about chords. Tangent segments lie outside the circle and are not chords.", "Correct. Draw radii to both points of tangency. Each radius is perpendicular to its tangent, the radii are congruent, and the two triangles share the hypotenuse from the centre to the external point — so they are congruent by HL, making the tangent segments congruent.", "Inscribed angles are not involved — nothing here has its vertex on the circle intercepting a shared arc.", "The two tangents intercept arcs of different sizes (a major and a minor arc), so this is not true, and it would not prove the segments congruent even if it were." ], "hint": "What can you draw from the centre to each point of tangency, and what does that create?", "solution": [ "Draw a radius to each point of tangency.", "Each radius is perpendicular to its own tangent, so both triangles are right-angled.", "The two radii are congruent, and both triangles share the hypotenuse from the centre to the external point.", "Hypotenuse-Leg makes the triangles congruent, so the tangent segments are congruent." ], "kind": "reasoning" }, { "id": "cg-049", "topic": "Reasoning: Error Analysis", "difficulty": "hard", "stem": "Two chords cross inside a circle, splitting one chord into 5 and 9 and the other into 3 and y. A student writes 5 + 9 = 3 + y and gets y = 11. What is the correct relationship and value?", "choices": [ "The differences are equal: 9 − 5 = y − 3, so y = 7.", "The PRODUCTS are equal: 5 × 9 = 3 × y, so y = 15.", "The chords must be congruent, so y = 9.", "The sums are equal; y = 11 is correct." ], "correctIndex": 1, "explanations": [ "Differences are not preserved. The relationship comes from similar triangles, which produce equal ratios and therefore equal products.", "Correct. When two chords intersect inside a circle, the products of the two pieces are equal: 5 × 9 = 45, so 3y = 45 and y = 15.", "Two chords crossing inside a circle are generally not congruent, and nothing here says these are.", "Equal sums would mean the two chords are the same length, which crossing chords need not be." ], "hint": "The relationship comes from a pair of similar triangles, so it produces a proportion — and a proportion cross-multiplies.", "solution": [ "Intersecting chords relate by PRODUCTS of their pieces, not by sums.", "5 × 9 = 3 × y", "45 = 3y", "y = 15" ], "kind": "reasoning" } ] } ], "id": "circle-geometry-staples" } } }