Question

in the figure M is the midpoint of QR angle p r q is equal to 90 degree prove that PQ square is equal to 4 p.m. square minus 3 P R square ​

Answer 1

Answer:

Step-by-step explanation:

Given :- In ∆PRQ, angle R = 90°

M is midpoint of RQ.

RM = MQ = 1/2 RQ

To Prove :- PQ^2 = 4PM^2 - 3PR^2

Proof :- In ∆PRM, angle PRM = 90°

By pythagoras theorem...,

PM^2 = PR^2 + RM^2

PM^2 = PR^2 + ( 1/2 RQ )^2 --( M is midpoint )

PM^2 = PR^2 + RQ^2/4

PM^2 = ( 4PR^2 + RQ^2 )/4

4PM^2 = 4PR^2 + RQ^2

RQ^2 = 4PM^2 - 4PR^2 --- ( 1 )

In ∆PRQ, angle PRQ = 90°

By pythagoras theorem..,

PQ^2 = PR^2 + RQ^2

PQ^2 = PR^2 + ( 4PM^2 - 4PR^2 ) --( From eq.1 )

PQ^2 = PR^2 + 4PM^2 - 4PR^2

PQ^2 = 4PM^2 - 3PR^2

Hence proved PQ^2 = 4PM^2 - 3PR^2