This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Equal Chords and their Distances from the Centre”.

1. What is the length of AB if CD = 8cm?

a) 5 cm

b) 8 cm

c) 4 cm

d) 2 cm

View Answer

Explanation: Chords which are equidistant from centre of a circle are equal in length. Hence, AB = CD = 8 cm.

2. Find the value of x if O is the centre of circle?

a) 20°

b) 45°

c) 90°

d) 40°

View Answer

Explanation: Equal chords subtend equal angles at the centre. As the chords are of same length of 3cm, x = 40°.

3. Find the value of ∠QOR if O is the centre of circle.

a) 50°

b) 65°

c) 130°

d) 30°

View Answer

Explanation: Join OP

In ΔPOQ, OP = OQ [Radii of same circle]

⇒ ∠OPQ = ∠OQP = 20° [Angles opposite to equal sides are equal]

In ΔPOR, OP = OR [Radii of same circle]

⇒ ∠OPR = ∠ORP = 45° [Angles opposite to equal sides are equal]

Now, ∠QPR = ∠OPR + ∠OPQ = 45° + 20° = 65°

Angle subtended by an arc of circle at the centre is twice the angle subtended by the arc on circumference ⇒ ∠ROQ = 2∠QPR = 2 x 65° = 130°.

4. Find the value of ∠ABC if O is the centre of circle.

a) 60°

b) 120°

c) 240°

d) 180°

View Answer

Explanation: Reflex ∠AOC = 240°

⇒ ∠AOC = 360° – Reflex ∠AOC = 360° – 240° = 120°

Since angle subtended by an arc of circle at the centre is twice the angle subtended by the arc on circumference, ∠AOC = 2∠ABC

⇒ ∠ABC = ½ ∠AOC = 60°.

5. What is the value of ∠PRQ if angle POQ and reflex angle POQ are in the ratio 1 : 3?

a) 135°

b) 50°

c) 115°

d) 80°

View Answer

Explanation: From figure, Angle around a point is 360° ⇒ ∠POQ + Reflex ∠POQ = 360°

⇒ x + 3x = 360° ⇒ x = 90°

Hence, Reflex ∠POQ = 3x = 270°

Since angle subtended by an arc of circle at the centre is twice the angle subtended by the arc on circumference, Reflex ∠POQ = 2∠PRB

⇒ ∠PRB = ½ x 270° = 135°.

6. Find the value of ∠PRQ from the given figure.

a) 50°

b) 160°

c) 80°

d) 30°

View Answer

Explanation: In ΔPSQ, ∠PSQ + ∠SPQ + ∠SQP = 180° [Angle sum property of triangle]

⇒ ∠PSQ + 50° + 50° = 180°

⇒ ∠PSQ = 80°

Angles subtended by an arc in the same segment are equal ⇒ ∠PRQ = ∠PSQ = 80°.

7. What is the value of ∠PRQ if ∠PTS : ∠TPS = 5 : 3?

a) 66°

b) 54°

c) 30°

d) 45°

View Answer

Explanation: From figure, ∠PTQ + ∠PTS = 180° [Linear Pair]

⇒ ∠PTS = 80°

In ΔPTS, ∠PST + ∠TPS + ∠PTS = 180° [Angle sum property of triangle]

⇒ ∠PST + 80° + 3/5 x 80° = 180° [∠PTS : ∠TPS = 5 : 3]

⇒ ∠PST = 54°

Since angles subtended by an arc in the same segment are equal, ∠PRQ = ∠PSQ = 54°.

8. Find the measure of arc PAQ.

a) 90°

b) 50°

c) 70°

d) 140°

View Answer

Explanation: Measure of an arc is angle subtended by it at the centre of circle.

⇒ m(PAQ) = ∠POQ

But, ∠POQ = 2∠PRQ [Angle subtended by an arc of circle at the centre is twice the angle subtended by the arc on circumference]

Hence, m(PAQ) = ∠POQ = 2∠PRQ = 140°.

9. What is the value of x if O is the centre of circle?

a) 90°

b) 110°

c) 70°

d) 55°

View Answer

Explanation: As angle subtended by an arc of circle at the centre is twice the angle subtended by the arc on circumference, ∠ABC = ½ reflex ∠AOC = ½ (360° – 110°) = 125°.

Now, ∠ABD + ∠ABC = 180° [Linear Pair]

⇒ x = 180° – 120° = 55°.

10. What is the value of ∠BDC if ∠BAC: ∠ABC = 4 : 5 and O is the centre of circle?

a) 40°

b) 10°

c) 90°

d) 55°

View Answer

Explanation: In ΔABC, ∠ACB = 90° as angle in a semicircle is right angle.

Also, ∠BAC = 4k and ∠ABC = 5k

Now, ∠ACB + ∠ABC + ∠BAC = 180° [Angle sum property of triangle]

⇒ 90° + 4k + 5k = 180°

⇒ k = 10°

⇒ ∠BAC = 4k = 40°

Since angles subtended by an arc in the same segment are equal, ∠BDC = ∠BAC = 40°.

11. Find the value of ∠QRS if RS || diameter PQ.

a) 90°

b) 15°

c) 40°

d) 20°

View Answer

Explanation: From figure, ∠PRQ = 90° as angle in a semicircle is right angle.

Also, ∠RPQ = ∠RTQ = 50° [angles subtended by an arc in the same segment are equal]

Now, in ΔPQR, ∠PRQ + ∠QPR + ∠PQR = 180° [Angle sum property of triangle]

⇒ 90° + 50° + ∠PQR = 180°

⇒ ∠PQR = 40°

Now, PQ || RS ⇒ ∠PQR = ∠QRS [Alternate interior angles]

⇒ ∠QRS = 40°.

12. If RS is equal to the radius of circle, find the value of ∠PTQ.

a) 90°

b) 60°

c) 45°

d) 30°

View Answer

Explanation: Join OS, OR and RQ

From figure, ∠PRQ = 90° as angle in a semicircle is right angle.

Now, ∠QRT = 180° – ∠PRQ [Linear Pair]

⇒ ∠QRT = 180° – 90° = 90° ————–(i)

In ΔROS, OS = OR = RS ⇒ ΔROS is an equilateral triangle.

⇒ ∠ROS = 60°

Now, ∠SQR = ½ ∠ROS [angle subtended by an arc of circle at the centre is twice the angle subtended by the arc on circumference]

⇒ ∠SQR = 30° —————-(ii)

In ΔTQR, ∠TRQ + ∠QTR + ∠TQR = 180° [Angle sum property of triangle]

⇒ 90° + ∠PQR + 30° = 180° [from equation i and ii]

⇒ ∠PQR = 60°.

**Sanfoundry Global Education & Learning Series – Mathematics – Class 9**.

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