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Q.

What will be the values of $∠ O P R, ∠ O P M$ and $∠ P R S$ , if in the given figure, PQ is the diameter of the circle having centre O with $∠ P Q R = 650, ∠ S P R = 400 & ∠ P Q M = 500$ ?

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a

25°, 40°, 25°

b

35°, 40°, 25°

c

25°, 40°, 45°

d

25°, 30°, 25°

answer is A.

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Detailed Solution

It is given that PQ is the diameter of the circle having centre O and,

∠ PQR=65 0, ∠ SPR=40 0, ∠ P Q M = ∠ 650

.

Question Image

According to the theorem, in a circle, the inscribed angle is half of the central angle.
We know that sum of all three angles of a triangle is

180^{°}

.
Also, if any four points on the circumference of a circle are joined, they form the vertices of a cyclic quadrilateral.
According to the property, the sum of the opposite angles of a cyclic quadrilateral is supplementary and the sum of all the angles equals 36

0^{°}

.
Calculating

∠ P R Q & ∠ P M Q

:
In a circle with centre O, POQ is a straight line.
Therefore, central angle

∠ P O Q = 1800

∠ P R Q & ∠ P M Q

are the inscribed angles.
Using the above theorem, we get,

∠ P R Q = 12 × 1800 = 900

……(1)
Likewise,

∠ P M Q = 900

Now

∠ Q P R & ∠ O P R

in ∆PQR,

∠ P R Q = 900

(from step 1)

∠ PQR=65 0

(given)
Using the property,

∠ Q P R + 900 + 650 = 1800 ⇒ ∠ Q P R = 1800 − 1550 ⇒ ∠ Q P R = 250

∠ Q P R

is the same as

∠ O P R

.
Therefore,

∠ O P R = 250

Calculating

∠ Q P M & ∠ O P M

in ∆QPM ,

∠ P M Q = 900

(from 1)

∠ P Q M = 500

(given)
Again using the property,

∠ Q P M + 900 + 500 = 1800 ⇒ ∠ Q P M = 1800 − 1400 ⇒ ∠ Q P M = 400

∠ Q P M

is the same as

∠ O P M

.
Therefore,

∠ O P M = 400

.
Since, all the four points P, Q, R and S lie on the circumference of a circle,
Therefore, PQRS is a cyclic quadrilateral.

⇒ ∠ P Q R + ∠ P S R = 1800 ⇒ ∠ P Q R = 650

So,

∠ P S R + 650 = 1800 ⇒ ∠ P S R = 1800 − 650 ⇒ ∠ P S R = 1150

Applying the property,

∠ P R S + ∠ P S R + ∠ R P S = 1800 ⇒ ∠ P R S = 1800 − 1550 ⇒ ∠ P R S = 250

So,

∠ O P R = 250, ∠ O P M = 400 & ∠ P R S = 250

.
Therefore, option 1 is correct.

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