Question

If the angle of intersection at a point where the two circles with radii $5cm$ and $12cm$ intersect is ${90}^{\circ },$ then the length $\left(\text{in}cm\right)$ of their common chord is :

• A. $\frac{120}{13}$
• B. $\frac{13}{2}$
• C. $\frac{60}{13}$
• D. $\frac{13}{5}$

Answer 1

The correct option is A $\frac{120}{13}$


Length of $C_1{C}_2=\sqrt{{12}^2+5^2}=13cm$
As both the radii intersect at ${90}^{\circ }\Rightarrow \angle A=\angle B=90°$
In $△A{C}_1{C}_2$
Area $\Delta =\frac{1}{2}\cdot A{C}_1\cdot A{C}_2=\frac{1}{2}\cdot 12\cdot 5=30cm$
Also, $\Delta =30=\frac{1}{2}\cdot AO\cdot 13$
$\Rightarrow AO=\frac{60}{13}cm$
$\therefore$ Length of common chord $AB=2\cdot AO=\frac{120}{13}cm$