Question

PQRS is a rectangle. PA = 16 cm, AQ = 9 cm and QR = 12 cm.

1. Find the lengths of AS and AR.
2. Prove that ∠SAR = 90°.

Answer 1

PQRS is a rectangle.

PA = 16, AQ = 9 cm

PQ = PA + AQ

= 16 + 9

= 25 cm

QR = 12 cm

PS = QR = 12 cm

SR = PQ = 25 cm

(i) Find the lengths of AS and AR.

In △APS (right-angled at P):

AS² = AP² + PS²

= 16² + 12²

= 256 + 144

= 400

AS = `sqrt400`

AS = 20 cm

In △AQR (right-angled at Q):

AR² = AQ² + QR²

= 9² + 12²

= 81 + 144

= 225

AR = `sqrt225`

AR = 15 cm

(ii) Prove that ∠SAR = 90°:

Consider △SAR: its sides are AS = 20, AR = 15, SR = 25.

Check Pythagoras theorem,

AS² + AR²

= 20² + 15²

= 400 + 225

= 625

SR² = 25² = 625

Since AS² + AR² = SR²

By the converse of Pythagoras theorem, triangle SAR is right-angled at A.

So, ∠SAR = 90°