Question

In circle, given below, $O$ is its centre and lengths of chords $AB$ and $CD$ are in the ratio $5:3$
If angle $AOB={100}^{\circ },$ find :
(i) $\angle COD$
(ii) ratio between the $arcAB$ and $arcCD$ .

Answer 1

REF.Image.

Given $\left(\frac{AB}{CD}\right)=\frac{5}{3}$ $\angle ADB={100}^{\circ }$

let AB = x

In $\Delta OAB$

$cos{100}^{\circ }=\frac{2r^2-x^2}{2r^2}$

$cos{100}^{\circ }=1-(\frac{x}{\sqrt{2r}}{)}^2$

$(\frac{x}{\sqrt{2r}}{)}^2=1-cos{100}^{\circ }$

$\frac{x^2}{2r^2}=2si{n}^2{50}^{\circ }$

$x=2rsin{50}^{\circ }$

r = 0.653x

(i) In $\Delta$ COD, cos $\angle$ COD $=1-\frac{C{D}^2}{2r^2}$

$cos\angle COD=1-\frac{9x^2}{25\left(2\right)(0.653{)}^2{x}^2}$

$cos\angle COD=0.5775$

$\angle COD={54.73}^{\circ }$

(ii) $\frac{\overline{AB}}{\overline{CD}}=\frac{\left(\frac{100\pi }{180}\right)\left(r\right)}{(54.73\frac{\pi }{180}{)}^r}$

$\left(\frac{\overline{AB}}{\overline{CD}}\right)=1.83$