Question

What is the perimeter of a segment whose end points subtend a ${60}^{\circ }$ angle at the center of a circle of diameter $3$ inches?

• A. $\frac{\pi }{2}+6$ inches
• B. $\frac{\pi }{2}+\frac{3}{2}$ inches
• C. $\frac{\pi }{2}+3$ inches

Answer 1

The correct option is B $\frac{\pi }{2}+\frac{3}{2}$ inches

The chord subtends a ${60}^{\circ }$ angle at the center of a circle. Since two of its sides that subtend this angle are equal to the radius, we can say that the triangle is isosceles. Hence the other two angles of the triangle must be equal.
From the angle sum property of a triangle, we can say that the sum of the other 2 angles must be ${180}^{\circ }-{60}^{\circ }={120}^{\circ }$.
Now, since both these angles are equal, each angle is ${60}^{\circ }$, making this triangle an equilateral triangle. Hence the length of the chord is equal to the radius and is equal to $\frac{3}{2}$ inches

Perimeter of the segment $=\frac{{60}^{\circ }}{{360}^{\circ }}\times 2\pi \left(\frac{3}{2}\right)+\frac{3}{2}$

$=\frac{\pi }{2}+\frac{3}{2}$