Circle Theorems Quiz

Grade 10 · Math · Quiz

Test your mastery of circle theorems including inscribed angles, chord properties, and cyclic quadrilaterals.

Questions

1. A central angle measures 84°. What is the measure of an inscribed angle that intercepts the same arc?

   1. 42°
   2. 84°
   3. 168°
   4. 21°
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   Answer: 42°

   An inscribed angle is half the central angle on the same arc: 84° ÷ 2 = 42°.

2. Points A, B, C, D lie on a circle in order. If angle ABC = 112°, what is angle ADC?

   1. 68°
   2. 112°
   3. 56°
   4. 124°
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   Answer: 68°

   ABCD is a cyclic quadrilateral, so opposite angles are supplementary: 180° − 112° = 68°.

3. From an external point P, two secants are drawn. The near and far intersections of the first secant are 3 and 8 from P. The near intersection of the second secant is 4 from P. How far is the far intersection of the second secant from P?

   1. 6
   2. 5
   3. 7
   4. 12
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   Answer: 6

   Power of a point: 3 × 8 = 4 × PB → PB = 24 ÷ 4 = 6.

4. A tangent from external point P touches the circle at T, and a secant from P passes through points A and B (A closer to P). If PT = 6 and PA = 4, what is PB?

   1. 9
   2. 6
   3. 8
   4. 12
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   Answer: 9

   Tangent-secant power: PT² = PA × PB → 36 = 4 × PB → PB = 9.

5. If a quadrilateral has one pair of opposite angles summing to 180°, it must be cyclic.

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   Answer: False

   Both pairs of opposite angles must each sum to 180° for a quadrilateral to be cyclic; one pair is not sufficient.

6. Two inscribed angles in the same circle intercept the same arc. Which statement is true?

   1. They are equal
   2. One is double the other
   3. They sum to 180°
   4. They are supplementary
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   Answer: They are equal

   Inscribed angles subtending the same arc are equal — both equal half the arc.

7. The perpendicular from the centre O of a circle to chord AB bisects AB. If OA = 10 and the perpendicular distance from O to AB is 6, what is half the chord length?

   1. 8
   2. 6
   3. 10
   4. 4
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   Answer: 8

   Using the Pythagorean theorem: half-chord² = OA² − d² = 100 − 36 = 64, so half-chord = 8.

8. The angle between a tangent to a circle and a radius drawn to the point of tangency is always ______ degrees.

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   Answer: 90

   A tangent is perpendicular to the radius at the point of tangency — this is the defining property of a tangent.

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