Question

Que: In Figure, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR.?​

Answer 1

Answer:

Solution:

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

So, the reflex ∠POR = 2 × ∠PQR

We know the values of angle PQR as 100°

So, ∠POR = 2 × 100° = 200°

∴ ∠POR = 360° – 200° = 160°

Now, in ΔOPR,

OP and OR are the radii of the circle

So, OP = OR

Also, ∠OPR = ∠ORP

Now, we know sum of the angles in a triangle is equal to 180 degrees

So,

∠POR + ∠OPR + ∠ORP = 180°

⇒ ∠OPR + ∠OPR = 180° – 160°

As ∠OPR = ∠ORP

⇒ 2∠OPR = 20°

Thus, ∠OPR = 10°

Hope it will be helpful :)

Answer 2

Answer:

$\huge\bold\red{REQUIRED \: ANSWER:-}$

Solution:

Take any point A on the circumcircle of the circle.

Join AP and AR.

∵ APQR is a cyclic quadrilateral.

∴ ∠PAR + ∠PQR = 180° [sum of opposite angles of a cyclic quad. is 180°]

∠PAR + 100° = 180°

⇒ Since ∠POR and ∠PAR are the angles subtended by an arc PR at the centre of the circle and circumcircle of the circle.

∠POR = 2∠PAR = 2 x 80° = 160°

∴ In APOR, we have OP = OR [radii of same circle]

∠OPR = ∠ORP [angles opposite to equal sides]

Now, ∠POR + ∠OPR + ∠ORP = 180°

⇒ 160° + ∠OPR + ∠OPR = 180°

⇒ 2∠OPR = 20°

⇒ ∠OPR = 10°

$\huge\boxed{\fcolorbox{blue}{orange}{hope it helps}}$