Question

In a right triangle with sides x and y hypotenuse z the altitude drawn on the hypotenuse is a then

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

Answer 1

The correct option is D $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{a^2}$

Given the sides of right angle triangle x and y and hypotenuse is z

Then $z^2=x^2+y^2$

Let the altitude on hypotenuse divided hypotenuse in side b and c

Then $x^2=a^2+b^2$......................(1)

and $y^2=a^2+c^2$.........................(2)

But b+c=z.........................................................(3)

Then square this equation

$b^2+2bc+c^2=z^2$..........................(4)

Add the equation (1) and (2)

$x^2+y^2=2a^2+b^2+c^2$

Or $x^2+y^2+2a^2=b^2+c^2$

But $z^2=x^2+y^2+2xy$......................(5)

comparing (4) and (5) we get

$a^2=xy$