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Harvard-MIT Math Tournament

Geometry Difficulty 4.1 AIME Find the answer United States

Problem:
A ladder is leaning against a house with its lower end 15 feet from the house. When the lower end is pulled 9 feet farther from the house, the upper end slides 13 feet down. How long is the ladder (in feet)?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
Of course the house makes a right angle with the ground, so we can use the Pythagorean theorem. Let $x$ be the length of the ladder and $y$ be the original height at which it touched the house. Then we are given
$x^{2} = 15^{2} + y^{2} = 24^{2} + (y - 13)^{2}.$
Isolating $y$ in the second equation we get $y = 20$ , thus $x$ is $25$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.