Question

The given diagram shows two isosceles triangles which are similar. In the given diagram, PQ and BC are not parallel; PC = 4, AQ = 3, QB = 12, BC = 15 and AP = PQ.

Calculate:

1. the length of AP,
2. the ratio of the areas of triangle APQ and triangle ABC.

Answer 1

i. Given, ΔAQP ~ ΔACB

`=> (AQ)/(AC) = (AP)/(AB)`

`=> (3)/(4 + AP) = (AP)/(3 + 12)`

`=>` AP² + 4AP – 45 = 0

`=>` (AP + 9)(AP – 5) = 0

`=>` AP = 5 units  ...(As length cannot be negative)

ii. Since, ΔAQP ~ ΔACB

∴ `(ar(ΔAPQ))/(ar(ΔACB)) = (PQ^2)/(BC^2)`

`=> (ar(ΔAPQ))/(ar(ΔABC)) = (AP^2)/(BC^2)`  ...(PQ = AP)

`=> (ar(ΔAPQ))/(ar(ΔABC)) = (5/15)^2 = 1/9`