Question 67·Medium·Geometry and Trigonometry

For a right triangle with two legs and a hypotenuse, what formula relates their lengths?

Which side is the hypotenuse (the longest side), and which sides are the legs in this situation?

After setting up $9^2 + h^2 = 15^2$, isolate $h^2$ and then take the square root to find $h$. Be careful not to subtract or add the side lengths directly without squaring.

In Desmos, type sqrt(15^2 - 9^2) to represent the height $h$ from the equation $9^2 + h^2 = 15^2$.

Look at the numeric value that Desmos displays for sqrt(15^2 - 9^2). That value is the height of the ladder on the wall; match it to the closest answer choice.

The ladder, the wall, and the ground form a right triangle:

• The ladder is the slanted side (the hypotenuse) with length 15 feet.
• The distance from the wall to the base of the ladder is one leg, 9 feet.
• The height up the wall where the ladder touches is the other leg; call this $h$.

For a right triangle with legs $a$ and $b$ and hypotenuse $c$, the Pythagorean theorem says:

Here, $a = 9$, $b = h$, and $c = 15$, so:

First compute the squares:

• $9^2 = 81$
• $15^2 = 225$

So the equation becomes:

Subtract 81 from both sides:

So the height squared is 144.

To find $h$, take the positive square root (height is a positive distance):

This gives $h = 12$ feet. The correct answer choice is 12.