Equilateral Triangles

Geometry Level 2

Let a a be the area of an equilateral triangle, and let b b be the area of another equilateral triangle inscribed in the incircle of the first triangle. What is a b \frac{a}{b} ?


The answer is 4.

L S by l s , 23, Canada

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2 solutions

After drawing the larger triangle and the incircle, join the three resulting points of tangency to form a triangle. By symmetry, this (inverted) inscribed triangle is also equilateral, and each side of this triangle is shared by another equilateral triangle within the larger triangle. As a result, the larger triangle has been divided into four congruent equilateral triangles, implying that a b = 4 \frac{a}{b} = \boxed{4} .

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Maybe you could give a more sound answer by means of calculations...

Aditya Singh - 4 years, 3 months ago
Syed Hissaan
Feb 21, 2017

as it can be seen that the area 1 is 1 / 4 1/4 of the total so answer is 4 / 1 = = > 4/1 ==> 4

3 Helpful 2 Interesting 0 Brilliant 0 Confused

Since the smaller equilateral triangle's circumradius will be equal to the larger equilateral traingle's inradius the side of the larger equilateral triangle will be in the ratio of 2:1 with the side of the smaller equilateral triangle (Because Circumradisu : Inradius is 2:1). Area of any equilateral triangle will be proportional to the square of its side so the ratio of Area of larger equilateral triangle (a) : Area of Smaller equilateral triangle (b) will be 2^2 : 1 i.e. 4:1

Nirat Patel - 1 year, 10 months ago

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