A ladder against a wall forms a right triangle with the ground. First identify the hypotenuse, the side opposite the right angle; the ladder takes that role. Then use the Pythagorean theorem or a familiar whole-number side pattern to find the height. Subtracting the two given lengths directly is tempting, but right-triangle side lengths are related by their squares.
Hints
- Hint 1
The hypotenuse is the side across from the right angle and is always the longest side. The wall and ground form that angle. Which given length is the hypotenuse?
- Hint 2
The Pythagorean theorem says the squares of the two legs add to the square of the hypotenuse. Call the wall height . Can you match the resulting equation to a familiar whole-number side pattern?
Step-by-step
Recognize the right-triangle side pattern
Step 1Identify the hypotenuse
No Desmos needed. These lengths match a familiar right-triangle side pattern. The wall and ground meet at the right angle. The -foot ladder is the hypotenuse, the side across from that angle. The height and the -foot ground distance are the legs, the two sides that meet at the right angle.
- Step 2
Relate the three sides
Call the height . The Pythagorean theorem says the legs' squares add to the hypotenuse's square, so:
- Step 3
Find the height
A Pythagorean triple is a set of three whole-number sides of a right triangle. The matching triple is , since . So , and the top of the ladder touches the wall feet above the ground. Choice C.