Math Core

Right triangles are hiding everywhere: a ladder leaning on a wall, the diagonal of a TV screen, a shortcut across a park. Once you spot the right triangle, the Pythagorean theorem finds the missing distance.

A plan for word problems

Most real-world problems follow the same four steps.

Solving a Pythagorean word problem

Draw a picture and find the right angle.
Label the two sides you know and the one you want. Decide which side is the hypotenuse (it's across from the right angle).
Write $a^2 + b^2 = c^2$ and fill in what you know.
Solve, then check that the answer makes sense and has units.

Ladders and walls

A wall meets the ground at a right angle. A ladder leaning against the wall is the hypotenuse, and the wall and the ground are the legs.

Worked example: How high does the ladder reach?

A $13$ -foot ladder leans against a wall. The bottom of the ladder is $5$ feet from the wall. How high up the wall does the ladder reach?

The wall and the ground are the legs. The ladder is the hypotenuse.

The ladder is the hypotenuse, so $c = 13$ . One leg is $5$ . Call the height $h$ .

\begin{aligned} 5^2 + h^2 &= 13^2 \\ 25 + h^2 &= 169 \\ h^2 &= 144 \\ h &= 12 \end{aligned}

The ladder reaches $12$ feet up the wall.

Diagonals of rectangles

A diagonal cuts a rectangle into two right triangles. The length and width are the legs, and the diagonal is the hypotenuse.

Worked example: The size of a TV

TV sizes are measured along the diagonal of the screen. A screen is $48$ inches wide and $27$ inches tall. What is its diagonal, to the nearest inch?

The diagonal splits the screen into two right triangles.

\begin{aligned} d^2 &= 48^2 + 27^2 \\ &= 2304 + 729 \\ &= 3033 \\ d &= \sqrt{3033} \approx 55.07 \end{aligned}

The screen is about $55$ inches, so it would be sold as a $55$ -inch TV.

Worked example: Taking a shortcut

A rectangular park is $120$ meters long and $50$ meters wide. Jordan usually walks along two edges to get to the opposite corner. How many meters does he save by cutting straight across on the diagonal?

Along the edges he walks $120 + 50 = 170$ meters.

The diagonal is $\sqrt{120^2 + 50^2} = \sqrt{14{,}400 + 2{,}500} = \sqrt{16{,}900} = 130$ meters.

He saves $170 - 130 = 40$ meters.

Common mistake

Decide whether the unknown is a leg or the hypotenuse before you calculate. In the ladder problem the ladder ( $13$ ) is the hypotenuse, so you subtract: $169 - 25$ . Adding instead gives $\sqrt{194} \approx 13.9$ feet, which is impossible: the ladder can't reach higher than its own length.

Right triangles in three dimensions

A box has a diagonal that goes from one bottom corner to the opposite top corner, straight through the inside. You can find it in two steps, using two right triangles.

Worked example: The longest pencil in a box

A box is $4$ inches wide, $3$ inches deep and $12$ inches tall. What is the longest straight pencil that fits inside, from a bottom corner to the opposite top corner?

Step 1: the diagonal of the bottom. The bottom is a $4$ by $3$ rectangle. Its diagonal is

$\sqrt{4^2 + 3^2} = \sqrt{25} = 5 \text{ inches}.$

Step 2: up to the top corner. Stand a right triangle up inside the box. One leg is the bottom diagonal ( $5$ inches). The other leg is the height of the box ( $12$ inches), which goes straight up, at a right angle to the bottom. The hypotenuse is the diagonal through the box:

$\sqrt{5^2 + 12^2} = \sqrt{169} = 13 \text{ inches}.$

The longest pencil that fits is $13$ inches.

Tip

When you have to round, keep the exact square root until the very last step. Rounding in the middle of a problem can make the final answer a little off.

Practice

Practice 1

A $10$ -foot ladder leans against a wall. Its base is $6$ feet from the wall. How high up the wall does the ladder reach, in feet?