Find the height of CH in the triangle ABC AB = BC = AC = 6√3.

Given:
triangle ABC,
AB = BC = AC = 6 √3,
CH – heights.
Find the length of the height CH -?
Solution:
Since AB = BC = AC = 6 √3, the triangle ABC is equilateral. Therefore, the height CH is also the median, then HB = 6 √3: 2 = 3 √3.
By the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):
AC ^ 2 + HC ^ 2 = BC ^ 2 (we express the legs BC ^ 2 from this equality);
HC ^ 2 = BC ^ 2 – HB ^ 2;
НС ^ 2 = (6 √3) ^ 2 – (3 √3) ^ 2;
CH ^ 2 = 108 – 27;
CH ^ 2 = 81;
CH = √81
CH = 9 centimeters – the length of the CH height.
Answer: CH = 9 centimeters.