Question

If ABC is an arc of a circle and $\angle ABC={135}^o$, then the ratio of arc $\stackrel{⌢}{ABC}$ to the circumference is

• A. 1 : 4
• B. 3 : 4
• C. 3 : 8
• D. 1 : 2

Answer 1

Answer: (B)

(1 : 4\)

Given $\angle ABC={135}^o$

Join OA and OC.

In cyclic quadrilateral, sum of opposite angles is ${180}^{\circ }$
$\Rightarrow \angle ABC+\angle ADC={180}^{\circ }$
$\Rightarrow {135}^{\circ }+\angle ADC={180}^{\circ }$
$\Rightarrow \angle ADC={180}^{\circ }-{135}^{\circ }$
$\therefore \angle ADC={45}^{\circ }$
$\Rightarrow \angle AOC=2\times \angle ADC$
$=2\times 45$
$\therefore \angle AOC={90}^{\circ }$
Hence, arc ABC represents quadrant of the circle.
So, the length of arc ABC=$\frac{1}{4}\times$ circumference of the circle.
Therefore, ratio of the length of arc ABC to the circumference is $1:4$