Question

In the given figure, P, Q and R are three points on

a circle such that the angles subtended by the

chords PQ and PR at the centre O are 80° and

120° respectively. Determine ∠QPR.​

Answer 1

Answer:

QPR=200°(PR120°+PQ80°)

Answer 2

Given:

Angle POQ=80°

Angle POR=120°

To find:

The angle QPR

Solution:

The measure of angle QPR is 80°.

We can find the measure by following the process given-

We know that PQ and PR are the chords of the circle.

Similarly, OQ, OP, and OR are the radii of the circle.

So, OQ=OP=OR.

Now, in triangle POQ,

OP=OQ (radii)

Angle OPQ=Angle OQP (Angles opposite to equal sides are also equal)

Also, Angle POQ+AngleOPQ+AngleOQP=180° (Sum of all angles)

Angle POQ=80°

On putting the values,

80°+ 2(Angle OPQ)=180°

2(Angle OPQ)=100°

Angle OPQ=50°

In a similar way, in triangle POR,

OP=OR (radii)

So, angle OPR=angle ORP (Angle opposite to equal sides are equal)

Angle POR+Angle OPR+Angle ORP=180°

Angle POR=120°

On putting the values,

120°+ 2(Angle OPR)=180°

2(Angle OPR)=60°

Angle OPR=30°

Now, angle QPR=angle OPR+angle OPQ

Putting the values,

Angle QPR=30°+50°=80°

Therefore, the measure of angle QPR is 80°.