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in Perimeter and Area of Plane Figures by (23.5k points)
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Find the ratio of the diameter of the circles inscribed in and circumscribing an equilateral triangle to its height?

(a) 1 : 2 : 1 

(b) 2 : 4 : 3 

(c) 1 : 3 : 4 

(d) 3 : 2 : 1

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(b) 2 : 4 : 3.

For an equilateral triangle of side a units,

In-radius = \(\frac{a}{2\sqrt3}\) units

⇒ Diameter of inscribed circle = \(\frac{a}{\sqrt3}\) units

Circumradius = \(\frac{a}{\sqrt3}\)

⇒ Diameter of circumscrible circle = \(\frac{2a}{\sqrt3}\) units

Height = \(\frac{\sqrt3}{2}a\) units.

∴ Required ratio = \(\frac{a}{\sqrt3}\) : \(\frac{2a}{\sqrt3}\)\(\frac{\sqrt3}{2}a\) = 2a : 4a : 3a = 2 : 4 : 3.

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