Question

answer the above attachment​

Answer 1

$\huge\bold{\gray{\sf{Answer:}}}$

$\bold{Explanation:}$

$\huge\bold{b)}$ $\huge\bold{45^{\circ}}$

Question:

If p,q,r are the lengths of the sides of a right triangle PQR and the hypotenuse r =$\bold{\sqrt{2pq}}$,then $\angle\:QPR =$_____

Given:

P,q,r are the lengths of the sides of a right triangle.

Hypotenuse r=$\bold{\sqrt{2pq}}$

To Find :

$\angle\:QPR=$______

$\sf r = \sqrt{2pq} ....(1)$

By Pythagoras theorem:

$\sf = > {p}^{2} + {q}^{2} = {r}^{2}$

$\sf= > {p}^{2} + {q}^{2} = 2pq$

$\sf= > \dfrac{ {p}^{2} }{pq} + \dfrac{ {q}^{2} }{pq} = 2$

$\sf= > \dfrac{p}{q} + \dfrac{q}{p} = 2$

Let $\bold{\dfrac{p}{q} \: be \: x}$

$\sf = > x + \dfrac{1}{x} = 2$

We get x=1

and

$\sf \dfrac{p}{q} = 1$

$\sf p = q$

✷We know that If any two sides of a right triangle are equal then their opposite angles are also equal.

$\bold{=>180^{\circ}-90^{\circ}=90^{\circ}}$

So, $\angle P=\angle R =45^{\circ}$