A -foot ladder leans against a vertical wall, forming a right triangle with the ground. The foot of the ladder is feet from the base of the wall. How many feet above the ground does the top of the ladder touch the wall?
For right-triangle ladder or building problems, quickly sketch the triangle and label each side: identify the hypotenuse (always opposite the right angle) and the legs. Apply the Pythagorean theorem , solving for the missing side, and be careful to square the lengths before adding or subtracting. On test day, also recognize common Pythagorean triples like –– so you can spot answers quickly without doing all the arithmetic.
Hints
Identify the triangle and its sides
Draw a quick sketch of the ladder leaning against the wall. Which side of the triangle is the hypotenuse, and which sides are the legs?
Use the Pythagorean theorem
You know the length of the hypotenuse and one leg of a right triangle. Which formula connects the legs and the hypotenuse of a right triangle?
Isolate the missing side
After writing , solve for by moving 25 to the other side, then take the square root to find .
Desmos Guide
Compute the missing side length
In Desmos, type the expression sqrt(13^2 - 5^2) and press Enter. This expression represents the length of the unknown leg found from .
Interpret the result
Look at the numerical output of sqrt(13^2 - 5^2); that value is the height, in feet, where the top of the ladder touches the wall.
Step-by-step Explanation
Visualize and label the right triangle
Picture the situation as a right triangle:
- The ground is one leg.
- The wall is the other leg (this is the height we want).
- The ladder is the hypotenuse (the side opposite the right angle).
Label the triangle:
- Hypotenuse (ladder): ft
- One leg (distance from wall): ft
- Other leg (height on wall): call it ft.
Set up the Pythagorean theorem
For a right triangle with legs and and hypotenuse , the Pythagorean theorem says:
Here, the hypotenuse is , and the legs are and , so:
Solve for the unknown height squared
Compute the squares and isolate :
Subtract 25 from both sides:
Take the square root to find the height
To get , take the square root of both sides:
Since a length is positive, .
So the top of the ladder touches the wall 12 feet above the ground, which corresponds to choice C.