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the circumference, not two points on the circumference. The required line segment is a chord."},{"text":"diameter","correct":false,"explanation":"A diameter passes through the centre, which is not required here. A line segment joining any two points on the circumference is a chord."},{"text":"tangent","correct":false,"explanation":"A tangent meets the circle at exactly one point rather than joining two points on it. The correct term is chord."}]},{"id":"95a1e60b-e7fa-2e5d-f47d-2db6c8e9b512","cardType":"cloze","points":null,"front":"A [ ] touches a circle at exactly one point.","back":"A tangent touches a circle at exactly one point.","mcqOptions":[{"text":"tangent","correct":true,"explanation":"A tangent is a straight line that meets a circle at exactly one point, called the point of contact."},{"text":"chord","correct":false,"explanation":"A chord is a line segment joining two points on the circumference. A line touching the circle at exactly one point is a tangent."},{"text":"secant","correct":false,"explanation":"A secant crosses the circle and meets its circumference at two points. A line meeting it at exactly one point is a tangent."},{"text":"radius","correct":false,"explanation":"A radius joins the centre of the circle to its circumference. The line that touches the circle at one point is a tangent."}]},{"id":"294aa9b1-2581-f9e5-9032-29fa2ea0d5ee","cardType":"basic","points":1,"front":"What fact lets you mark OA and OB equal when O is the centre and A and B lie on the circle?","back":"All radii in the same circle are equal.","mcqOptions":[{"text":"All radii in the same circle are equal.","correct":true,"explanation":"OA and OB both run from the centre O to the circumference, so both are radii of the same circle and must have equal lengths."},{"text":"All chords in the same circle are equal.","correct":false,"explanation":"Chords in the same circle can have different lengths depending on their distance from the centre. OA and OB are equal because they are radii."},{"text":"All radii in different circles are equal.","correct":false,"explanation":"Radii are equal only when they belong to the same circle; different circles may have different sizes. OA and OB are radii of one circle."},{"text":"All lines from the centre are equal.","correct":false,"explanation":"Only segments from the centre to the circumference are necessarily radii. OA and OB are equal specifically because both are radii of the same circle."}]},{"id":"b2b2d872-2c50-dcdc-b1b4-f3bb70cacc17","cardType":"basic","points":1,"front":"If two sides of a triangle are radii of the same circle, what type of triangle is it?","back":"An isosceles triangle.","mcqOptions":[{"text":"An isosceles triangle.","correct":true,"explanation":"Two radii of the same circle have equal lengths, so the triangle has two equal sides and is therefore an isosceles triangle."},{"text":"An equilateral triangle.","correct":false,"explanation":"Two equal radii guarantee only two equal sides, not all three. The triangle must be isosceles, but it is not necessarily equilateral."},{"text":"A scalene triangle.","correct":false,"explanation":"A scalene triangle has three unequal sides, which contradicts the fact that the two radii are equal. The triangle is isosceles."},{"text":"A right-angled triangle.","correct":false,"explanation":"Equal radii do not guarantee a 90° angle. They create two equal sides, so the triangle is isosceles rather than necessarily right-angled."}]},{"id":"85fda3a4-0ecb-5be5-9b2f-080850600ff1","cardType":"basic","points":1,"front":"In triangle AOB, OA=OB and ∠ABO=42°. What is ∠OAB?","back":"42°.","mcqOptions":[{"text":"42°.","correct":true,"explanation":"OA and OB are equal radii, making AOB isosceles. Its base angles at A and B are therefore equal, so ∠OAB is 42°."},{"text":"48°.","correct":false,"explanation":"Subtracting 42° from 90° incorrectly assumes the triangle is right-angled. Since OA and OB are equal, both base angles are 42°."},{"text":"84°.","correct":false,"explanation":"Doubling the given base angle does not find the other base angle. In an isosceles triangle the base angles are equal, so ∠OAB is 42°."},{"text":"96°.","correct":false,"explanation":"This is the angle at O after subtracting both 42° base angles from 180°. The requested base angle ∠OAB is 42°."}]},{"id":"b0b072cd-0bad-ea7d-2b63-dab212f36494","cardType":"basic","points":1,"front":"In triangle AOB, the base angles are 42° and 42°. What is ∠AOB?","back":"96°.","mcqOptions":[{"text":"96°.","correct":true,"explanation":"Angles in a triangle total 180°. Subtracting both base angles gives $180°-42°-42°=96°$, so ∠AOB is 96°."},{"text":"84°.","correct":false,"explanation":"Adding the two base angles gives 84°, but the question asks for the remaining angle. Subtracting 84° from 180° gives 96°."},{"text":"138°.","correct":false,"explanation":"Subtracting only one 42° angle overlooks the second base angle. Both must be removed from 180°, giving ∠AOB as 96°."},{"text":"42°.","correct":false,"explanation":"The equal base angles are each 42°, but the angle at O is different. Triangle angles total 180°, so ∠AOB is 96°."}]},{"id":"d7a9cc94-5e2b-dc4a-670a-7f3eb8068e45","cardType":"cloze","points":null,"front":"A tangent to a circle is [ ] to the radius at the point of contact.","back":"A tangent to a circle is perpendicular to the radius at the point of contact.","mcqOptions":[{"text":"perpendicular","correct":true,"explanation":"At the point of contact, a tangent and the radius drawn to that point meet at a right angle, so they are perpendicular."},{"text":"parallel","correct":false,"explanation":"A tangent cannot be parallel to the radius at their shared point of contact. They meet at 90°, so the correct relationship is perpendicular."},{"text":"oblique","correct":false,"explanation":"An oblique line meets another at an angle other than 90°. The tangent-radius theorem states that these lines are perpendicular at the contact point."},{"text":"equal in length","correct":false,"explanation":"The theorem concerns the angle between the tangent and radius, not their lengths. They are perpendicular at the point of contact."}]},{"id":"d04b3e43-2489-b136-fd5b-68e6bc1b2b05","cardType":"basic","points":1,"front":"A tangent touches the circle at A and OA is a radius. What is the angle between OA and the tangent?","back":"90°.","mcqOptions":[{"text":"90°.","correct":true,"explanation":"A tangent is perpendicular to the radius at the point of contact. Perpendicular lines meet at a right angle, which measures 90°."},{"text":"45°.","correct":false,"explanation":"An angle of 45° is half a right angle and does not satisfy the tangent-radius theorem. OA and the tangent meet at 90°."},{"text":"180°.","correct":false,"explanation":"An angle of 180° would place OA and the tangent on one straight line. A tangent is perpendicular to the radius, giving 90°."},{"text":"0°.","correct":false,"explanation":"An angle of 0° would mean OA and the tangent point in the same direction. They are perpendicular at A, so the angle is 90°."}]},{"id":"75ce04c3-9796-6988-61d5-13371b0dd6b7","cardType":"basic","points":1,"front":"A tangent touches the circle at A. Which radius makes the 90° angle with the tangent?","back":"The radius to the point of contact.","mcqOptions":[{"text":"The radius to the point of contact.","correct":true,"explanation":"A tangent is perpendicular to the radius drawn to the point where it touches the circle, so this radius forms the 90° angle."},{"text":"The radius to the opposite point.","correct":false,"explanation":"The radius to the opposite side lies on the same diameter, but it does not meet the tangent. The perpendicular radius ends at the point of contact."},{"text":"The radius parallel to the tangent.","correct":false,"explanation":"Parallel lines do not form a 90° angle. The radius drawn to the point of contact is perpendicular to the tangent."},{"text":"The radius to the centre of the chord.","correct":false,"explanation":"A radius may be perpendicular to a chord at its midpoint, but the stated line is a tangent. The relevant radius goes to the point of contact."}]},{"id":"eedfcb22-8dc0-e4f3-03c0-9ef4760ca032","cardType":"cloze","points":null,"front":"If two angles stand on the same chord or arc, the angle at the centre is [ ] the angle at the circumference.","back":"If two angles stand on the same chord or arc, the angle at the centre is twice the angle at the circumference.","mcqOptions":[{"text":"twice","correct":true,"explanation":"For angles subtended by the same chord or arc, the angle at the centre is twice the corresponding angle at the circumference."},{"text":"half","correct":false,"explanation":"This reverses the circle theorem. The circumference angle is half the centre angle, so the angle at the centre is twice the angle at the circumference."},{"text":"equal to","correct":false,"explanation":"Equal angles arise in the same segment, not when comparing centre and circumference angles. The centre angle is twice the corresponding circumference angle."},{"text":"three times","correct":false,"explanation":"The multiplier in the centre-angle theorem is 2, not 3. The angle at the centre is twice the angle at the circumference."}]},{"id":"12cd42a2-fde9-3f47-e6f6-a724797a8f82","cardType":"basic","points":1,"front":"∠BOC at the centre is 74°. ∠BAC at the circumference stands on the same chord BC. What is ∠BAC?","back":"37°.","mcqOptions":[{"text":"37°.","correct":true,"explanation":"The angle at the circumference is half the angle at the centre when both stand on chord BC, so $74° \\div 2 = 37°$."},{"text":"74°.","correct":false,"explanation":"This incorrectly treats the centre and circumference angles as equal. The circumference angle is half of 74°, giving the correct answer of 37°."},{"text":"148°.","correct":false,"explanation":"This doubles the centre angle instead of halving it. Since the circumference angle is half the 74° centre angle, the correct answer is 37°."},{"text":"53°.","correct":false,"explanation":"This results from subtracting from 180°, which is not the relevant theorem. Halving the 74° centre angle gives the circumference angle of 37°."}]},{"id":"deee6846-5c82-c027-20db-7a6d8edc75b8","cardType":"basic","points":1,"front":"A reflex centre angle is 220°. What is the smaller centre angle?","back":"360° - 220° = 140°.","mcqOptions":[{"text":"360° - 220° = 140°.","correct":true,"explanation":"Angles around the centre total 360°. Subtracting the 220° reflex angle leaves the smaller centre angle: $360° - 220° = 140°$."},{"text":"220° - 180° = 40°.","correct":false,"explanation":"Subtracting a straight angle does not find the remaining angle around a point. The two centre angles total 360°, so the smaller angle is 140°."},{"text":"360° - 220° = 180°.","correct":false,"explanation":"The correct method is shown, but the subtraction is inaccurate. Since $360 - 220 = 140$, the smaller centre angle is 140°."},{"text":"360° - 140° = 220°.","correct":false,"explanation":"This calculation returns the given reflex angle rather than the requested smaller angle. Subtracting 220° from 360° gives the smaller angle of 140°."}]},{"id":"79a25ff2-81c7-b28c-143f-716729eb89b4","cardType":"cloze","points":null,"front":"Angles in the [ ] are equal.","back":"Angles in the same segment are equal.","mcqOptions":[{"text":"same segment","correct":true,"explanation":"Angles subtended by the same chord at points on the same side of that chord are equal, which is the same-segment circle theorem."},{"text":"same sector","correct":false,"explanation":"Being within the same sector does not guarantee equal angles. The theorem applies to angles standing on the same chord in the same segment."},{"text":"same circle","correct":false,"explanation":"Angles located in the same circle can have many different sizes. They are equal when they stand on the same chord in the same segment."},{"text":"opposite segment","correct":false,"explanation":"Angles on opposite sides of a chord are generally supplementary rather than equal. The equal-angle theorem applies to angles in the same segment."}]},{"id":"07e4a359-5a3d-55da-6886-1bcd9e727b9d","cardType":"basic","points":1,"front":"Angles ADB and ACB both stand on chord AB in the same segment. If ∠ADB=47°, what is ∠ACB?","back":"47°.","mcqOptions":[{"text":"47°.","correct":true,"explanation":"Angles ADB and ACB stand on the same chord AB in the same segment, so the same-segment theorem makes them equal at 47°."},{"text":"133°.","correct":false,"explanation":"This supplements 47° to 180°, confusing the situation with opposite angles in a cyclic quadrilateral. Same-segment angles are equal, so ∠ACB is 47°."},{"text":"94°.","correct":false,"explanation":"This doubles 47° using the centre-angle theorem, but neither stated angle is at the centre. Same-segment angles are equal, so ∠ACB is 47°."},{"text":"23.5°.","correct":false,"explanation":"This halves 47° as though one angle were at the centre. Both angles are at the circumference in the same segment, so ∠ACB is 47°."}]},{"id":"f1b06190-16db-e5c0-cb7b-37436133be14","cardType":"cloze","points":null,"front":"Opposite angles in a cyclic quadrilateral add to [ ].","back":"Opposite angles in a cyclic quadrilateral add to 180°.","mcqOptions":[{"text":"180°","correct":true,"explanation":"A quadrilateral whose four vertices lie on a circle is cyclic, and each pair of opposite interior angles is supplementary, adding to 180°."},{"text":"360°","correct":false,"explanation":"The four interior angles of any quadrilateral total 360°, but one opposite pair in a cyclic quadrilateral adds to 180°."},{"text":"90°","correct":false,"explanation":"An angle in a semicircle is 90°, but that is a different circle theorem. Opposite angles in a cyclic quadrilateral add to 180°."},{"text":"270°","correct":false,"explanation":"This confuses the supplementary relationship with a reflex total. Each pair of opposite angles in a cyclic quadrilateral adds to 180°, not 270°."}]},{"id":"ddb6a712-7250-ce8d-e27f-f22425913d11","cardType":"basic","points":1,"front":"ABCD is cyclic and ∠ADC=78°. What is the opposite angle ∠ABC?","back":"102°.","mcqOptions":[{"text":"102°.","correct":true,"explanation":"Opposite angles in a cyclic quadrilateral add to 180°, so $\\angle ABC = 180° - 78° = 102°$."},{"text":"78°.","correct":false,"explanation":"This incorrectly treats opposite cyclic angles as equal. They are supplementary, so subtracting 78° from 180° gives ∠ABC as 102°."},{"text":"282°.","correct":false,"explanation":"This subtracts 78° from 360°, using the total around a point. Opposite cyclic-quadrilateral angles total 180°, giving ∠ABC as 102°."},{"text":"156°.","correct":false,"explanation":"This doubles 78°, confusing the problem with the centre-angle theorem. Opposite angles in a cyclic quadrilateral sum to 180°, so ∠ABC is 102°."}]},{"id":"55f61fbb-378b-5e87-7d56-92bea55f961c","cardType":"cloze","points":null,"front":"The [ ] links a tangent-chord angle to the angle in the opposite segment.","back":"The alternate segment theorem links a tangent-chord angle to the angle in the opposite segment.","mcqOptions":[{"text":"alternate segment theorem","correct":true,"explanation":"The alternate segment theorem states that the angle between a tangent and a chord equals the angle subtended by that chord in the opposite segment."},{"text":"same segment theorem","correct":false,"explanation":"The same segment theorem compares two angles subtended by the same chord within a circle. The tangent-chord relationship is given by the alternate segment theorem."},{"text":"angle at the centre theorem","correct":false,"explanation":"The angle at the centre theorem relates central and circumferential angles subtended by the same arc. A tangent-chord angle uses the alternate segment theorem."},{"text":"cyclic quadrilateral theorem","correct":false,"explanation":"The cyclic quadrilateral theorem states that opposite angles in a cyclic quadrilateral sum to 180°. The tangent-chord relationship is the alternate segment theorem."}]},{"id":"38dbea96-e04d-917c-4588-ac9fd6ee74c2","cardType":"basic","points":1,"front":"In triangle ABC, ∠ABC=58° and ∠ACB=71°. What is ∠BAC?","back":"51°.","mcqOptions":[{"text":"51°.","correct":true,"explanation":"Angles in a triangle sum to 180°, so ∠BAC = 180° - 58° - 71° = 51°."},{"text":"49°.","correct":false,"explanation":"This results from an arithmetic slip when subtracting the two known angles. Since triangle angles total 180°, ∠BAC = 180° - 58° - 71° = 51°."},{"text":"109°.","correct":false,"explanation":"Adding 58° and 51° does not find the remaining interior angle. The two given angles total 129°, leaving ∠BAC = 51°."},{"text":"129°.","correct":false,"explanation":"This is the sum of the two given angles, not the third angle. Subtracting 129° from the triangle total of 180° gives ∠BAC = 51°."}]},{"id":"1feca021-0a2f-957e-0fd7-e74129df90ca","cardType":"basic","points":1,"front":"At C, a tangent angle uses chord CB. In triangle ABC, which angle is in the opposite segment?","back":"∠BAC.","mcqOptions":[{"text":"∠BAC.","correct":true,"explanation":"Chord CB subtends the angle at A on the opposite circumference, so the angle in the opposite segment is ∠BAC."},{"text":"∠ABC.","correct":false,"explanation":"This angle has vertex B, one endpoint of chord CB, so it is not subtended by chord CB. The opposite-segment angle has vertex A and is ∠BAC."},{"text":"∠ACB.","correct":false,"explanation":"This angle has vertex C, where the tangent meets the chord, rather than in the opposite segment. Chord CB subtends ∠BAC at A."},{"text":"∠BOC.","correct":false,"explanation":"This is a central angle rather than an angle at the circumference in the opposite segment. The angle subtended by chord CB at A is ∠BAC."}]},{"id":"c970abe5-ea4d-5e35-4e2f-47cd399adb23","cardType":"basic","points":1,"front":"A tangent makes angle x with chord CB, and the opposite-segment angle ∠BAC is 51°. What is x?","back":"51°.","mcqOptions":[{"text":"51°.","correct":true,"explanation":"By the alternate segment theorem, the angle between the tangent and chord CB equals the angle subtended by chord CB in the opposite segment, so x = 51°."},{"text":"39°.","correct":false,"explanation":"Subtracting from 90° wrongly assumes the tangent-chord angle is complementary to ∠BAC. The alternate segment theorem makes the two angles equal, so x = 51°."},{"text":"129°.","correct":false,"explanation":"Subtracting 51° from 180° gives the supplementary exterior angle, not the stated tangent-chord angle. The alternate segment theorem gives x = 51°."},{"text":"102°.","correct":false,"explanation":"Doubling 51° confuses this theorem with the rule that an angle at the centre is twice an angle at the circumference. Here, x = 51°."}]},{"id":"dc44283f-01f8-a2f2-0f24-44951dc89c99","cardType":"basic","points":2,"front":"In right-angled triangle OAC, OA=6 cm and AC=8 cm. What is OC?","back":"10 cm.","mcqOptions":[{"text":"10 cm.","correct":true,"explanation":"OC is the hypotenuse, so Pythagoras’ theorem gives $OC=\\sqrt{6^2+8^2}=\\sqrt{100}=10$ cm."},{"text":"14 cm.","correct":false,"explanation":"Adding the two shorter sides directly does not calculate a hypotenuse. Pythagoras’ theorem gives $OC=\\sqrt{6^2+8^2}=10$ cm."},{"text":"7 cm.","correct":false,"explanation":"Averaging the two shorter sides is not valid for finding the hypotenuse. Using Pythagoras’ theorem gives $OC=\\sqrt{36+64}=10$ cm."},{"text":"12 cm.","correct":false,"explanation":"This can result from mishandling the squared side lengths. The correct calculation is $OC=\\sqrt{6^2+8^2}=\\sqrt{100}=10$ cm."}]},{"id":"20d1c5ba-0f50-2e4e-9cc9-c1ad6cb5cff7","cardType":"basic","points":1,"front":"O, B and C are collinear, with OC=10 cm and OB=6 cm. What is BC?","back":"4 cm.","mcqOptions":[{"text":"4 cm.","correct":true,"explanation":"B lies between O and C, so $OC=OB+BC$. Therefore $BC=10-6=4$ cm."},{"text":"16 cm.","correct":false,"explanation":"Adding OC and OB ignores that OB is already part of the full distance OC. Since B lies between O and C, $BC=10-6=4$ cm."},{"text":"5 cm.","correct":false,"explanation":"Halving OC would only work if B were the midpoint, which is not stated and conflicts with $OB=6$ cm. Subtraction gives $BC=4$ cm."},{"text":"6 cm.","correct":false,"explanation":"This repeats the length OB rather than finding the remaining section of OC. Since $OC=OB+BC$, $BC=10-6=4$ cm."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Circle Theorems","title":"Circle Theorems Flashcards","topbar":{"backHref":"/gcse/maths/ocr/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"8208bc9f-e8a1-4a0e-b129-e7b3c45edef1","walled":"$L39"}],"$L3a","$L3b"]