Questions: A ladder is 13 feet long and leans up against the side of a house. The house and the ground form a right angle. The bottom of the ladder is 5 feet from the house. What is the distance between the ground and the top of the ladder, in feet?

Solution Steps

We can represent the situation with a right triangle where the ladder is the hypotenuse, the distance from the house to the bottom of the ladder is one leg, and the distance from the ground to the top of the ladder is the other leg.

Let \(x\) be the distance between the ground and the top of the ladder. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs). In this case, the ladder is the hypotenuse with length 13 feet, one leg has length 5 feet, and the other leg has length \(x\) feet. Therefore, we have \(13^2 = 5^2 + x^2\) \(169 = 25 + x^2\)

Subtract 25 from both sides: \(169 - 25 = x^2\) \(144 = x^2\) Take the square root of both sides: \(x = \sqrt{144}\) \(x = 12\)

Final Answer