CAT 2023 Slot 2 QA Question 17: n a rectangle $ABCD$, $AB = 9$ cm and $BC = 6$ cm. $P$ and $Q$ are two points on $BC$ such that the areas of the figures $ABP$, $APQ$, and $AQCD$ are in geometric progression. If the area of the figur

Question

n a rectangle $ABCD$, $AB = 9$ cm and $BC = 6$ cm. $P$ and $Q$ are two points on $BC$ such that the areas of the figures $ABP$, $APQ$, and $AQCD$ are in geometric progression. If the area of the figur

Accepted Answer

Let $BP = x$, $PQ = y$ and $QC = z$. Then $x + y + z = 6$.

The areas of triangles $ABP$ and $APQ$ have the same height $AB = 9$ cm.
$	ext{Area}(ABP) = \dfrac{9x}{2}$
$	ext{Area}(APQ) = \dfrac{9y}{2}$

The third area is four times the first, and the three areas are in geometric progression. Hence,