A 13-foot-long ladder leans against the side of a building. The bottom of the ladder is 5 feet from the base of the building. What height does the ladder reach?

5^2 + h^2 = 13^2

h^2 = 169 - 25 = 144
h = 12 remarkably

You can save yourself some calculation by learning a few of the basic Pythagorean triples, such as

3-4-5, 5-12-13, 8-15-17, 7-24-25
and their multiples.

To find the height the ladder reaches, we can use the Pythagorean Theorem. According to the theorem, in a right triangle, the sum of the squares of the lengths of the two shorter sides (legs) is equal to the square of the length of the longest side (hypotenuse).

In this scenario, the ladder acts as the hypotenuse, and the base of the building and the height it reaches form the two legs of the triangle.

Given that the bottom of the ladder is 5 feet from the base of the building, this length represents one of the legs of the triangle. The length of the ladder is the hypotenuse, which is 13 feet.

To find the height, we will let 'h' represent the unknown height.

Using the Pythagorean Theorem, we can form the equation:

(Length of one leg)^2 + (Length of the other leg)^2 = (hypotenuse)^2
5^2 + h^2 = 13^2
25 + h^2 = 169
h^2 = 169 - 25
h^2 = 144

To solve for 'h', we can take the square root (√) of both sides:

√(h^2) = √144
h = 12

Therefore, the ladder reaches a height of 12 feet.