CAT 2023 Slot 1 QA Question 19: In a right-angled triangle ∆ABC, the altitude AB is 5 cm, and the base BC is 12 cm. P and Q are two points on BC such that the areas of ∆ABP, ∆ABQ and ∆ABC are in arithmetic progression. If the area o

Question

In a right-angled triangle ∆ABC, the altitude AB is 5 cm, and the base BC is 12 cm. P and Q are two points on BC such that the areas of ∆ABP, ∆ABQ and ∆ABC are in arithmetic progression. If the area o

Accepted Answer

The area of ∆ABC is
$\dfrac{1}{2} 	imes 5 	imes 12 = 30$ sq. cm.

Given that this is 1.5 times the area of ∆ABP,
$	ext{Area}(ABP) = \dfrac{30}{1.5} = 20$

Since P lies on BC, the altitude from A to BC is the same 5 cm.
$	ext{Area}(ABP) = \dfrac{1}{2} 	imes 5 	imes BP = 20$
$BP = 8$

The areas

In a right-angled triangle ∆ABC, the altitude AB is 5 cm, and the base BC is 12 cm. P and Q are two points on BC such that the areas of ∆ABP, ∆ABQ and ∆ABC are in arithmetic progression. If the area of ∆ABC is 1.5 times the area of ∆ABP, the length of PQ, in cm, is