Question

If the altitude $AD$ of an acute angled $\Delta ABC$ meets the opposite side $BC$ at $D$ and the circumcircle at $E$, then the length of $DE$ is

• A. $\frac{a\sin B\sin C}{\cos A}$
• B. $\frac{a\cos B\cos C}{\sin A}$
• C. $\frac{a\sin A}{\cos B\cos C}$
• D. $\frac{a}{\cos B\cos C}$

Answer 1

The correct option is B $\frac{a\cos B\cos C}{\sin A}$

Angle made by chord $AB$ at $E$ and $C$ are equal.

Also angle made by chord $AC$ at $B$ and $E$ are equal.

In $\Delta BDE$;

$\tan C=\frac{C\cos B}{DE}⟹DE=CcosB.\cos C$

$⟹DE=C\cos B.\frac{\cos C}{\sin C}$

$⟹DE=\frac{a.\cos B.\cos C}{\sin A}$

[Since using Sine law:$\frac{a}{\sin A}=\frac{C}{\sin C}$]