Question

pqr is an equilateral triangle and ps is perpendicular to qr,prove that 4ps²= 3pr²​

Answer 1

Answer:

Given, PS⊥QR

Let PS=2x,QS=4x,RS=x

Now, in △PSR

PR

2

=PS

2

+RS

2

PR

2

=4x

2

+x

2

PR=

5

x

In △PQS

PQ

2

=PS

2

+QS

2

PQ

2

=4x

2

+16x

2

PQ=

20

x

Now, PQ

2

+PR

2

=20x

2

+5x

2

=25x

2

and, PS

2

=(4x+x)

2

=25x

2

Thus, PQ

2

+PR

2

=PS

2

By converse of Pythagoras theorem, the triangle is a right angled triangle.

Answered By

Answer 2

Answer:

Given, PS⊥QR

Let PS=2x,QS=4x,RS=x

Now, in △PSR

PR

2

=PS

2

+RS

2

PR

2

=4x

2

+x

2

PR=

5

x

In △PQS

PQ

2

=PS

2

+QS

2

PQ

2

=4x

2

+16x

2

PQ=

20

x

Now, PQ

2

+PR

2

=20x

2

+5x

2

=25x

2

and, PS

2

=(4x+x)

2

=25x

2

Thus, PQ

2

+PR

2

=PS

2

By converse of Pythagoras theorem, the triangle is a right angled triangle.