Question

In ΔАВС, ∠ACB = 90°. P and Q are the points on CA and CB respectively which divides these sides in the ratio 2 : 1. Then 9(AQ² + BP²) is equal to ______.

Options

• 5AB²

• 8AB²

• 10AB²

• 13AB²

Answer 1

In ΔАВС, ∠ACB = 90°. P and Q are the points on CA and CB respectively which divides these sides in the ratio 2 : 1. Then 9(AQ² + BP²) is equal to 13AB².

Explanation:

Place C at (0, 0), A at (a, 0) and B at (0, b) so AB² = a² + b². 

If P and Q divide CA and CB in the ratio 2 : 1 measured from C.

Then `P = ((2a)/3,0)` and `Q = (0, (2b)/3)`.

Using the distance formula Pythagoras, we get

`AQ^2 = a^2 + ((2b)/3)^2`

= `(9a^2 + 4b^2)/9`

`BP^2 = ((2a)/3)^2 + b^2`

= `(4a^2 + 9b^2)/9`

So, AQ² + BP² 

= `(13a^2 + 13b^2)/9` 

= `13(a^2 + b^2)/9` 

Therefore, 9(AQ² + BP²)

= 13(a² + b²)

= 13AB²