Mathematics Questions and Answers – Equal Chords and their Distances from the Centre

«
»

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

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

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

a) 20°
b) 45°
c) 90°
d) 40°
View Answer

Answer: d
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

Answer: c
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°.
advertisement
advertisement

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

a) 60°
b) 120°
c) 240°
d) 180°
View Answer

Answer: a
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

Answer: a
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°.
advertisement

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

a) 50°
b) 160°
c) 80°
d) 30°
View Answer

Answer: c
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

Answer: b
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°.
advertisement

8. Find the measure of arc PAQ.

a) 90°
b) 50°
c) 70°
d) 140°
View Answer

Answer: d
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

Answer: d
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°.
advertisement

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

Answer: a
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

Answer: c
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

Answer: b
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.

To practice all areas of Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers.

Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs!

advertisement
advertisement
Manish Bhojasia - Founder & CTO at Sanfoundry
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter