Question

In a right angled triangle, if the square of the hypotenuse is twice the product of the other two sides, then one of the angles of the triangle is:

• A. ${15}^o$
• B. ${30}^o$
• C. ${45}^o$
• D. ${60}^o$

Answer 1

The correct option is C ${45}^o$

Given,
$A{C}^2=AB\times BC$
Also,by Pythogoras Theorem , $\angle B=90$
$A{C}^2=A{B}^2+A{C}^2$
$\Rightarrow 2AB\times BC=A{B}^2+A{C}^2$
$\Rightarrow A{B}^2+A{C}^2-2AB\times BC=0$
$\Rightarrow A{B}^2+A{C}^2-2AB\times BC=0$
$\Rightarrow (AB-AC{)}^2=0$
$\Rightarrow AB=AC$
Thus,the opposite angles are equal.
$\Rightarrow \angle ACB=\angle CAB$------(i)
Now, in $△ABC$
$\angle ABC+\angle ACB+\angle CAB=180$
$\Rightarrow 90+\angle ACB+\angle ACB=180$ [ from (i)]
$\Rightarrow 2\angle ACB=180-90$
$\Rightarrow \angle ACB=\frac{90}{2}=45$
Therefore,
one of the angle is $45$