Question

13. What is the approximate sum of the radii of the circumcircle and incircle of an equilateral triangle
of side v3 units?
(a) 1.5 units
(b) 2 units
(c) 2.5 units
(d) 3 units​

Answer 1

Answer:

Option (a) 1.5 units

Step-by-step explanation:

If ABC is a triangle with sides a, b and c then

The relation between the Area A side lengths a, b, c; circumradius R and inradius r is

$R=\frac{abc}{4A}$

$A=rs$       where s is semi-perimeter of the triangle

Given the length of the side of the equilateral traingle = $\sqrt{3}$

Area A of the traingle

$A=\frac{\sqrt{3}}{4} \times \text{side}^2$

$\implies A=\frac{\sqrt{3}}{4} \times (\sqrt{3} )^2$

$\implies A=\frac{3\sqrt{3}}{4}$ unit²

∴ The radius of the circumcircle

$R=\frac{\sqrt{3}\times \sqrt{3}\ \times \sqrt{3}\ }{4A}$

$\implies R=\frac{3\sqrt{3}}{4\times\frac{3\sqrt{3} }{4}}$

$\implies R=1$ unit

The radius of the incircle

$r=\frac{A}{s}$

$\implies r=\frac{3\sqrt{3}/4 }{3\sqrt{3}/2}$

$\implies r=1/2$ unit

Therefore,

$R+r=1+1/2=3/2$ units

or, R + r = 1.5 units