Sequences & Series
Infinite GP — Perimeters and Areas of Similar Triangles
nta_pyq_2024_apr
Grade 11
Question:
Let $ABC$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $ABC$ and the same process is repeated infinitely many times. If $P$ is the sum of perimeters and $Q$ is the sum of areas of all the triangles formed in this process, then:
$P^2=6\sqrt{3}Q$
$P^2=36\sqrt{3}Q$
$P=36\sqrt{3}Q^2$
$P^2=72\sqrt{3}Q$
Step-by-Step Solution
Key Concept: Let side of ABC $=a$. Perimeters form GP: $3a,3a/2,\ldots$ Sum $P=6a$. Areas form GP: $\frac{\sqrt{3}}{4}a^2,\frac{\sqrt{3}}{4}\cdot\frac{a^2}{4},\ldots$ Sum $Q=\frac{\sqrt{3}a^2}{3}$.
$P=6a$, $Q=\sqrt{3}a^2/3$. $P^2=36a^2=36\sqrt{3}Q$.
Correct Answer: 2