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CAT 2018 Slot 1 — QA Question 32

Similarity of TrianglesEasy

Given an equilateral triangle T1 with side 24 cm, a second triangle T2 is formed by joining the midpoints of the sides of T1. Then a third triangle T3 is formed by joining the midpoints of the sides of T2. If this process of forming triangles is continued, the sum of the areas, in sq cm, of infinitely many such triangles T1, T2, T3,... will be

Answer & solution

  • A

    248√3

  • B

    164√3

  • C

    188√3

  • 192√3

Solution

Consider ∆ABC (i.e. T1) and ∆DEF (i.e., T2).

D and E are midpoints of AB and AC respectively. Therefore, BC = 2 × DE

Side of T2 = 1/2 × Side of T1

Area of T1 = A(T1) = √3/4 × 242

Similarly, A(T2) = √3/4 × 122

A(T3) = √3/4 × 62 ... and so on

Sum of areas of infinitely many Ti’s

= √3/4 × 242 + √3/4 × 122 + √3/4 × 62 + ...

= √3/4 (242 + 122 + 62 + ...)

Here, (242 + 122 + 62 + ...) is an infinite series with r = 1/4

Hence, = √3/4 (242 + 122 + 62 + ...) = 34(242(1-14)) = 192√3

Hence, option (d).

CAT 2018 Slot 1 QA Q32: Given an equilateral triangle T 1 with side 24 cm, a second triangle T 2 is formed by joining the midpoints of — Solution | TheCATExam