Question

An equilateral triangle of side 9 cm is inscribed a circle. The radius of the circle is (a) 3 cm (b) 3v2 cm (c) 3v3 cm (d) 6 cm​

Answer 1

Answer:

Step-by-step explanation:

Consider △ ABC as an equilateral triangle with side 9 cm

Take AD as one of its medians

We know that

AD ⊥ BC

It can be written as

BD = ½ × BC

By substituting the values

BD = ½ × 9

So we get

BD = 4.5 cm

Consider △ ADB

Using the Pythagoras theorem

AB^2 = AD^2 + BD^2

Substituting the values

9^2 = AD^2 + (9/2)^2

On further calculation

AD^2 = 9^2 – (9/2)^2

So we get

AD^2 = 81 – 81/4

By taking out the square root

AD = 9√3/ 2 cm

We know that the centroid and circumcenter coincide in an equilateral triangle

AG: GD = 2: 1

The radius can be written as

AG = 2/3 AD

By substituting the values

AG = (2/3) × (9√3/ 2)

So we get

AG = 3√3 cm

Therefore, the radius of the circle is 3√3 cm.

therefore option c is correct