Question

hello!
In the given circle O is the centre.
angle PQR=49°
and
angle QRS=21°
Find angle OPS​

Answer 1

Answer:

answer should be /_ops=8°

Answer 2

Concept:

The angle of a circle is 360°

We know that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

If two sides are equal , then the two angles opposite to the equal sides will also be equal.

Given:

The value of the following angles are given

∠SRO=21°

∠PQO=49°

To find:

The value of ∠OPS

Solution:

Lets join OS and form line OS

From ΔORS we can understand that

∠ORS=∠OSR       [ Since OR=OS as the are the radius of the circle]

∴∠OSR=21°               ---------------(1)

Since we know that angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Therefore,

2∠RQP=∠ROP     [ Arc RTP ]

⇒∠ROP=2×49

⇒∠ROP=98°

∴∠ROP=98°

Also, using the same theorem

2∠RSP=∠ROP

⇒∠RSP=$\frac{1}{2}$∠ROP

⇒∠RSP=$\frac{1}{2}(98)$

⇒∠RSP=49°              --------------(2)

Now we know that

∠OSR+∠OSP=∠RSP

From equation 1 and 2

⇒21+∠OSP=49

⇒∠OSP=49-21

⇒∠OSP=28°            --------------(3)

From ΔOPS we can understand that

∠OPS=∠OSP       [ Since OP=OS as the are the radius of the circle]

∴∠OPS=28°          [  From equation (3) ]

Therefore, the value of ∠OPS is 28° .