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:

POQ+POR+QOR = 360 (a circle is 360°)

80+120+QOR = 360

200+QOR = 360

QOR = 360-200

QOR = 160

QOR = 2×QPR (Angle subtended by arc at the centre is double the angle subtended by it at any other point)

160 = 2×QPR

160÷2 = QPR

QPR = 80°

Answer 2

Given:

P, Q, R are the three points on the circle.

Two chords PQ and QR are subtended at the center 80° and 120°respectively.

To Find:

∠QPR

Solution:

In the given figure,

∠POQ = 80° and ∠POR = 120°

The angle subtended at the center is doubled the angle at circumference by the same chord..(i)

As we know a circle os 360°

So,

⇒ ∠POQ + ∠POR + ∠QOR = 360°

Put the measures of the following angles to find ∠QOR.

⇒ 80° + 120° + ∠QOR = 360°

⇒ 200° + ∠QOR = 360°

Now, we subtract 200 from 360,

⇒ ∠QOR = 360° - 200°

⇒ ∠QOR = 160°

∠QOR = 2×∠QOR

[Angle subtended at the center is doubled the angle at circumference by the same chord]

Using (i),

⇒ 160° = 2×∠QPR

We put the values and multiply them by 2 to get the measure of angle QPR.

⇒ 160/2 = ∠QPR

∠QPR = 80°.

Therefore, the measure of ∠QPR = 80°.