Question

In a right triangle with sides a and b and hypotenuse c,the altitude drawn on the hypotenuse is x. Then:

• A. a.b= x${}^2$
• B. $\frac{1}{a}+\frac{1}{b}=\frac{1}{x}$
• C. $a^2+b^2=2x^2$
• D. $\frac{1}{x^2}=\frac{1}{a^2}+\frac{1}{b^2}$
• E. $\frac{1}{x}=\frac{b}{a}$

Answer 1

Answer: (D)

$\frac{1}{x^2}=\frac{1}{a^2}+\frac{1}{b^2}$

The segments of the hypotenuse are
$\frac{a^2}{c}$ and $\frac{b^2}{c}$.Using similar triangles,$\frac{x}{a}=\frac{\frac{b^2}{c}}{b},\frac{x}{b}=\frac{\frac{a^2}{c}}{a}$.
$\therefore x^2=\frac{a^2}{c}\frac{b^2}{c}=\frac{a^2.b^2}{a^2+b^2}$; and $\therefore \frac{1}{x^2}=\frac{1}{a^2}+\frac{1}{b^2}$.