Question

The length of a side of an equilateral triangle is 12. What is the length of the altitude in simplest radical form?

Answer 1

`6sqrt(3)`

Explanation:

Construct a line segment which intersects one vertex of the triangle and forms a `90^@`angle with the opposite side. In an equilateral triangle, this bisects the opposite side, and gives the following diagram.

Note that this line segment is also the altitude of the triangle, so we just need to solve for `x`. To do so, we use the Pythagorean theorem.

`6^2 + x^2 = 12^2 => x = sqrt(12^2-6^2) = sqrt(108)`
(note that as `x>0` there is no need to consider negative roots)

Finally, we simplify the result.

`108 = 2^2*3^3=>sqrt(108)=sqrt((2*3)^2*3)=6sqrt(3)`

Thus `x = 6sqrt(3)`