Question

In a right angled triangle, one acute angle is double the other angle. Which of these statements is correct?

• A. Hypotenuse is double the shorter side
• B. Hypotenuse is half the shorter side
• C. Hypotenuse is double the longer side
• D. Hypotenuse is half the longer side

Answer 1

The correct option is A Hypotenuse is double the shorter side

It is given that, in a right angled triangle, one acute angle is double the other angle.


So, let $\angle C=2\angle A$.
Now, let’s construct AD = AC which meets CB produced at C.

In $\Delta ABC$ and $\Delta ABD,$
$\angle ABC=\angle ABD={90}^0$
$AC=AD$ (by consruction)
$AB=AB$ (common)
$\Delta ABC\cong \Delta ABD,$ (by RHS congruence criterion)
$BD=BC$ (CPCT)........(1)
$\angle BDA=\angle BCA$ .........(2)
$\angle BAD=\angle BAC$ .........(3)
$\therefore \angle C=2\angle A=2x\left(let\right)$
$\angle BDA=\angle BCA$ (from(2))
$\Rightarrow \angle BDA=2x$
and $\angle BAD=\angle BAC$ (from(3))
$\Rightarrow \angle BAD=x$
In $\Delta ADC,$
$\angle ADC=\angle ACD=\angle DAC=2x$
So, $\Delta ADC$ is equilateral triangle
$AD=AC=DC$
$AC=DC\Rightarrow AC=BD+DC$
$\Rightarrow AC=2BC$ (from (i))
So, the hypotenuse is double the shorter side.
Thus, the correct answer is option (a).