CAT 2025 Slot 2 QA Question 15: Let ABCDEF be a regular hexagon and P and Q be the midpoints of AB and CD, respectively. Then, the ratio of the areas of trapezium PBCQ and hexagon ABCDEF is

Question

Let ABCDEF be a regular hexagon and P and Q be the midpoints of AB and CD, respectively. Then, the ratio of the areas of trapezium PBCQ and hexagon ABCDEF is

Accepted Answer

Let the side length of the regular hexagon be $a$.

The area of the regular hexagon is
$\dfrac{3 \sqrt{3} a^2}{2}$

Since P and Q are midpoints of AB and CD, the parallel sides of trapezium PBCQ are
$BC = a$
$PQ = \dfrac{3a}{2}$
and the height between them is $\dfrac{\sqrt{3}a}{4}$.

$A_{PBCQ} = \df