A 15-foot ladder is leaning against a vertical wall so that its base is feet from the foot of the wall. How many feet above the ground does the top of the ladder touch the wall?

Solution available with step-by-step explanation

SAT Right Triangles & Trigonometry Question 17 | Free SAT Prep | Aniko

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Question 17·Medium·Right Triangles and Trigonometry

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A 15-foot ladder is leaning against a vertical wall so that its base is $9$ feet from the foot of the wall. How many feet above the ground does the top of the ladder touch the wall?

For ladder and wall problems, immediately recognize the right triangle setup and identify which side is the hypotenuse (the ladder) versus the legs (height and distance from the wall). Write the Pythagorean equation with the hypotenuse as the side alone on one side of the equation, plug in the known values, and simplify systematically: square the numbers, isolate the unknown squared term, then take the positive square root. Avoid common traps like just adding or subtracting the side lengths without squaring.

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Identify the triangle

Think about the shape formed by the wall, the ground, and the ladder. What kind of triangle is it?

Choose the correct theorem

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

Assign the sides correctly

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

Solve the equation carefully

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.

Desmos Guide

Enter the Pythagorean expression

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

Read the numerical result

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.

Step-by-step Explanation

Visualize the right triangle

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 $h$ .

Set up the Pythagorean theorem

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

a^2 + b^2 = c^2

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

9^2 + h^2 = 15^2

Solve for the square of the height

First compute the squares:

$9^2 = 81$
$15^2 = 225$

So the equation becomes:

81 + h^2 = 225

Subtract 81 from both sides:

h^2 = 225 - 81 = 144

So the height squared is 144.

Find the height and match the answer choice

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

h = \sqrt{144}

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

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation