The bisector of an equilateral triangle is 9√3. find his side.

Let us denote by a cm the side of an equilateral triangle. In an equilateral triangle, the bisector is also the height, i.e. the bisector divides the given triangle into two rectangular ones. In the resulting right-angled triangle, one side (equal to a cm) will be the hypotenuse, the bisector will be the leg, the other leg will be half the side, i.e. a / 2. Then, by the Pythagorean theorem:

a ^ 2 = (a / 2) ^ 2 + (9√3) ^ 2

a ^ 2 = a ^ 2/4 + 27 (multiply both sides of the equation by 4)

4a ^ 2 = a ^ 2 + 108

3a ^ 2 = 108

a ^ 2 = 108/3

a ^ 2 = 36

a = 6.

Answer: the side of the triangle is 6 cm.