Question

the shortest distance AP from a point A to a straight line QR is 12 cm and Q,R are 15 cm and 20 cm distance from A on opposite sides of AP. Prove that QAR is a right angle.

Answer 1

Δ AQR is right angled. (Proved)

Step-by-step explanation:

We have AP = 12 cm, AQ = 15 cm and AR = 20 cm.

Now, Δ APR is a right triangle as AP ⊥ QR and AP is the shortest distance from point A to line QR.

So, AQ² = AP² + PQ²

⇒ 15² = 12² + PQ²

⇒ PQ = 9 cm.

Again, Δ APR is a right triangle and AR² = AP² + PR²

⇒ 20² = 12² + PR²

⇒ PR = 16 cm.

Therefore, QR = QP + PR = 9 + 16 = 25 cm.

Now, Δ AQR has AQ = 15 cm, QR = 25 cm and RA = 20 cm and

QR² = AQ² + RA² as 25² = 15² + 20²

So, as the Pythagoras Theorem is satisfied, hence, Δ AQR is right angled. (Proved)