Question

Write your question here (Keep it simple and clear to get the best answer)the radius of the circumcircle of an equilateral triangle is 20cm.find the length of each side of the equilateral triangle

Answer 1

FIGURE IS ATTACHED 

Here we have an equilateral ∆ ABC and a circumcircle with center O.

OA, OB and OC are radius = 20 cm

Radius is equidistant from the center, so O is circumcenter (meeting point of perpendicular bisectors)

If we extend BO, CO and AO it becomes perpendicular bisectors.

We know that in an equilateral triangle,

Altitudes, Medians, Angle bisectors lie on same line. So AD, BF and CE are medians and perpendicular bisectors.

Take ∆ OBD which is a right ∆

OB = 20 cm

BD = DC (AD is median)

Since, O is centroid (meeting point of medians), it divides median into 2 parts such that one part is double the other. Vertex to center is longer part.

AO = 2 x OD

20 = 2 x OD

OD = 10 cm

By, Pythagoras theorem

OB² = OD² + BD²

20² = 10² + BD²

BD = 10√3 cm

BC = 2 X 10√3 = 20√3 cm.

Therefore, length of each side = 20√3 cm.

OR

Each angle of an equilateral triangle = 60°

So, ∠OBD = 30° (BF is angle bisector)

∆ OBD is a right ∆

Cos 30° = base/hypotenuse

√3/2 = BD / OB

√3/2 = BD/20

BD = 10√3cm

BC = 2 x 10√3 = 20√3 cm