Question

In a right triangle ABC right angled at C, P and Q are points on the sides CA and CB respectively, which divide these sides in the ratio 2:1. Then, which of the following statements is true?

(a) 9AQ² = 9 BC² + 4 AC²

(b) 9AQ² = 9 AC² + 4 BC²

(c) 9AQ² = 9 BC² + 4 AC²

(d) 9AQ² = 9 AB² – 4 BP²

Answer 1

Correct option is (b) 9AQ² = 9 AC² + 4 BC²

It is given that P divides CA in the ratio 2 : 1. Therefore,

CP = \(\frac 23\) AC   ……(1)

Also, Q divides CB in the ratio 2 : 1

∴ QC = \(\frac 23\) BC    .......(2)

Applying Pythagoras theorem in right-angled triangle ACQ, we have

AQ² = QC² + AC²

⇒ AQ² = \(\frac 49\) BC² + AC²   [Using (2)]

⇒ 9 AQ² = 4 BC² + 9 AC²