The length of a side of an equilateral triangle is 4. Which type of number best describes the length of the altitude of the triangle?

To find the length of the altitude of an equilateral triangle, we can use the Pythagorean theorem.

Step 1: We know that an equilateral triangle has all sides of equal length. So, the base of the triangle is also 4.

Step 2: We draw the altitude from one vertex to the midpoint of the opposite side. This divides the equilateral triangle into two right-angled triangles.

Step 3: In each of these right-angled triangles, the hypotenuse is the side of the equilateral triangle (which is 4), one side is half the base of the equilateral triangle (which is 4/2 = 2), and the other side is the altitude (which we are trying to find).

Step 4: According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, we have:

4^2 = 2^2 + altitude^2

16 = 4 + altitude^2

altitude^2 = 16 - 4 = 12

altitude = sqrt(12) = 2sqrt(3)

So, the length of the altitude of the triangle is 2sqrt(3), which is an irrational number. Therefore, the type of number that best describes the length of the altitude of the triangle is an irrational number.