Question

If the arcs of the same length in two circles subtend angles ${45}^{\circ }$ and ${135}^{\circ }$ at the centre, find the ratio of their radii.

• A. $3:1$
• B. $2:1$
• C. $1:3$
• D. $1:2$

Answer 1

The correct option is A $3:1$

Let $r_1$ and $r_2$ be the radii of the two circles.
Given that,
${\theta }_1={45}^{\circ }=\frac{\pi }{4}\text{rad}$
${\theta }_2={135}^{\circ }=\frac{3\pi }{4}\text{rad}$
Let $l$ be the length of each of the arc, then,
$l=r_1{\theta }_1=r_2{\theta }_2$, which gives

$r_1\frac{\pi }{4}=r_2\frac{3\pi }{4}\Rightarrow \frac{r_1}{r_2}=\frac{\frac{3\pi }{4}}{\frac{\pi }{4}}=\frac{3}{1}$
Hence, $r_1:r_2=3:1$