The bisector of an equilateral triangle is 9√3. Find its side.

Given:

ABC – equilateral triangle,

ВO – bisector,

BO = 9√3.

Find the lengths of the sides of the triangle ABC -?

Solution:

1. Consider an equilateral triangle ABC. In it AB = AC = BC. The ВO bisector is both the height and the median. Then OС = AO = 1/2 * AC.

2. Consider a right-angled triangle ABO. Let AO = x centimeters and AB = 2x centimeters. Then, by the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):

AO ^ 2 + BO ^ 2 = AB ^ 2:

x ^ 2 + (9√3) ^ 2 = (2x) ^ 2;

x ^ 2 + 243 = 4x ^ 2;

4x ^ 2 – x ^ 2 = 243;

3 * x ^ 2 = 243;

x ^ 2 = 243: 3;

x ^ 2 = 81;

x = 9 is the length of the AO;

9 * 2 = 18 – length AB.

Answer: 18.