If the angles of a triangle are 450, 450, and 900, then each of the perpendicular sides is (1/√2 ) times the hypotenuse.

If the angles of a triangle are 45⁰, 45⁰, and 90⁰, then each of the perpendicular sides is (1/√2 ) times the hypotenuse.

Given:

In ∆ PQR, ∠ P = 45⁰, ∠ Q = 90⁰, ∠ R = 45⁰.

To Prove	:	PQ	=	QR	=	1	PR
						√2

Solution: In ∆ PQR,

∠ P ≅ ∠ R

∴ It in an Isosceles triangle

∴ QR = PQ   ---------------------- eq. no. 1

[Converse of Isosceles triangle theorem]

Now, In right angled ∆ PQR,

PR² = PQ² + QR²  [By Pythagoras theorem]

∴ PR² = PQ² + PQ²  [From eq. no. (1)]

∴ PR² = 2PQ²

Taking root on both the sides,

∴ PR = √2 PQ

PQ	=	1	PR
		√2

∴	PQ	=	QR	=	1	PR	From eq (1)
					√2