Question

If sum of angle subtended by arc at center and at circumference is ${108}^0$, then find the angle at center and at circumference.

Answer 1

In circle $C(O,r)$ angle subtended by arc $PQ$ at center $\angle POQ$ and internal angle in alternate segment is $\angle PRQ$
$\angle POQ+\angle PRQ={108}^0$
We know that angle subtended by arc at center of circle is double the angle at remaining part.
$\angle POQ=2\angle PRQ$ …..(i)
But $\angle POQ+\angle PRQ={108}^0$
$2\angle PRQ+\angle PRQ={108}^0$
$3\angle PRQ={108}^0$
$\angle PRQ=\frac{{108}^0}{2}={36}^0$
$\angle PRQ$ putting value of in equation (i)
$\angle POQ=2\angle PRQ=2\ast 36={72}^0$
Angle at center of circle = ${72}^0$ and angle of remaining part = ${36}^0$.