if perimeter of a certain sector of a circle is equal to half of the circle of which it is - Maths - Circles

Question

if perimeter of a certain sector of a circle is equal to half of
the circle of which it is a part then circular measure of the angle
of sector
2- plz draw fig and then solve this
two concentric circle has been drawn with centre o a right
angled triangle inside the circle in such a way that hypotenuse
touches the smaller circle as a tangent of of smaller circle and
perpendicular is drawn as the radius of bigger circle and the base
is also the radius of bigger circle find the radius of smaller
circle

Lalit Mehra · Accepted Answer

Let r be the radius of the circle and Î¸ be the angle subtended by the arc of the corresponding sector at the centre Given, perimeter of sector $= \frac{1}{2} × Circumfence of the circle$ $\Rightarrow \frac{θ}{360 °} \times 2 π r + 2 r = \frac{1}{2} \times 2 π r \Rightarrow \frac{θ}{360 °} \times 2 π + 2 = π \Rightarrow \frac{θ}{360 °} \times 2 π = π - 2 \Rightarrow \frac{θ}{180 °} = 1 - \frac{2}{π} = 1 - \frac{14}{22} = \frac{8}{22} \Rightarrow θ = \frac{8}{22} \times 180 ° \Rightarrow θ = 65 5 ° (Approx)$