JEE Mathematics 2024 Question with Solution

Given: An equilateral triangle of side $a$ is repeatedly replaced by the triangle formed by joining the midpoints of its sides.

Find: The relation between $P$ and $Q$.

The side lengths of successive triangles are:

So their perimeters form a geometric series:

Using the sum of the infinite geometric series,

The area of an equilateral triangle of side $a$ is:

Hence the areas of successive triangles are:

Therefore,

Again using the sum of the infinite geometric series,

Now from

we get

Substituting into $Q$,

So,

Therefore, the correct option is A.

Given: Each new triangle is obtained by joining the midpoints of the sides of the previous equilateral triangle.

Find: A direct relation between $P^2$ and $Q$.

By the midpoint theorem, each new triangle has side half of the previous one. Therefore the perimeters are in the ratio $1: \frac{1}{2}: \frac{1}{4}: \dots$ and the areas are in the ratio $1: \frac{1}{4}: \frac{1}{16}: \dots$.

Perimeter sum:

Hence,

Area sum:

Since

we get

Therefore,

Hence,

This matches option A. Note: the line in the source solution stating "Answer: $(1)\; P = 36\sqrt{3}Q$" is a typo; the derived result and the option list both confirm $P^2 = 36\sqrt{3}Q$.