Lesson 3: Apply the Pythagorean Theorem to Solve Problems

Property

The Pythagorean theorem states that for a right triangle with legs of length $a$ and $b$ and a hypotenuse of length $c$, the relationship is $a^2 + b^2 = c^2$.

This theorem is used to find an unknown side length in any right triangle when the other two side lengths are known. It is widely applied in real-world problems involving distances and heights.

Examples

• What is the length of the diagonal of a rectangular TV screen that is 12 inches tall and 16 inches wide? The diagonal is the hypotenuse, so its length is $\sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20$ inches.
• A 17-foot ladder is placed against a wall. The base of the ladder is 8 feet away from the wall. How high up the wall does the ladder reach? Let the height be $h$. Then $h^2 + 8^2 = 17^2$, so $h^2 = 289 - 64 = 225$. The ladder reaches a height of $h = \sqrt{225} = 15$ feet.
• A ship sails 9 miles north and then 12 miles east. How far is the ship from its starting point? The path forms a right triangle. The distance $d$ is $\sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$ miles.

Property

To find the length of the diagonal $d$ that passes through the interior of a rectangular prism (a box) with length $l$, width $w$, and height $h$, we can use a three-dimensional version of the Pythagorean theorem. The formula is:

This is derived by applying the Pythagorean theorem twice: first for the diagonal of the base, and then for the triangle formed by the base diagonal, the height, and the interior diagonal.

Examples

• What is the length of the longest fishing pole that can fit in a box that is 3 feet wide, 4 feet long, and 12 feet tall? The length is $\sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$ feet.
• What is the length of the diagonal inside a cube with a side length of 2 cm? The length is $\sqrt{2^2 + 2^2 + 2^2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3}$ cm.
• A room is 12 feet long, 16 feet wide, and 9 feet high. The diagonal on the floor is $\sqrt{12^2+16^2} = \sqrt{144+256} = \sqrt{400} = 20$ feet. The space diagonal is then $\sqrt{20^2 + 9^2} = \sqrt{400+81} = \sqrt{481}$ feet.

Explanation

To find the longest straight line inside a box, you use the Pythagorean theorem in 3D. Imagine finding the diagonal on the floor, then using that line and the box's height to form a new right triangle that cuts through the box's center.

Property

The Pythagorean theorem, $a^2 + b^2 = c^2$, can be used to find unknown lengths in real-world scenarios that can be modeled by a right triangle. In the formula, $a$ and $b$ represent the lengths of the legs, and $c$ represents the length of the hypotenuse.

Examples