Let A B be a diameter of a circle and let C be a point on the...

Let

A B

be a diameter of a circle and let

C

be a point on the segment

A B

such that

A C: C B=6: 7

. Let

D

be a point on the circle such that

D C

is perpendicular to

A B

. Let

D E

be the diameter through

D

. If

[X Y Z]

denotes the area of the triangle

X Y Z

. Find

[A B D] /[C D E]

to the nearest integer. Let A B be a diameter of a circle and let C be a point on the segment A B such that A C: C B=6: 7 . Let D be a point on

Correct answer

\begin{aligned} & C D=\sqrt{r^2-\frac{r^2}{13^2}}=\frac{r}{13} \sqrt{12 \times 14} \\ & \tan \alpha=\frac{1}{\sqrt{12 \times 14}} \Rightarrow \sin \alpha=\frac{1}{\sqrt{12 \times 14+1}} \\ & \frac{C P}{C D}=\sin \alpha=\frac{C P}{\frac{r}{13} \sqrt{12 \times 14}}=\frac{1}{\sqrt{12 \times 14+1}} \\ & C P=\frac{r \sqrt{12 \times 14}}{13 \sqrt{12 \times 14+1}} \\ & \frac{|A B D|}{|C D E|}=\frac{\frac{1}{2}(2 r) \times \frac{r}{13} \sqrt{12 \times 14}}{\frac{1}{2}(2 r) \times \frac{r}{13} \frac{\sqrt{12 \times 14}}{\sqrt{12 \times 14+1}}}=\sqrt{12 \times 14+1}=13\end{aligned}