Pythagorean theorem word problems: 3 worked examples

The ladder, the wall, and the ground form a right-angled triangle where the ladder is the hypotenuse, and the distance from the base of the wall and the height up the wall are the two legs. Let $c$ be the length of the ladder ($13$ feet), $a$ be the distance from the base of the wall ($5$ feet), and $b$ be the height the ladder reaches on the wall. According to the Pythagorean theorem, $a^{2} + b^{2} = c^{2}$. Substituting the known values, we get $5^{2} + b^{2} = 13^{2}$. This simplifies to $25 + b^{2} = 169$. Subtracting $25$ from both sides gives $b^{2} = 144$. Taking the square root of both sides, we find $b = \sqrt{144} = 12$. Thus, the ladder reaches $12$ feet up the wall.

A rectangular field with length $l = 40$ meters and width $w = 30$ meters forms a right-angled triangle when divided by a diagonal path. The diagonal $d$ acts as the hypotenuse of this triangle. According to the Pythagorean theorem, $a^{2} + b^{2} = c^{2}$, where $a$ and $b$ are the sides and $c$ is the hypotenuse. Substituting the given values, we get $d^{2} = 40^{2} + 30^{2}$. Calculating the squares, $d^{2} = 1600 + 900$, which simplifies to $d^{2} = 2500$. Taking the square root of both sides, $d = \sqrt{2500} = 50$. Therefore, the length of the diagonal path is $50$ meters.

The boat's path forms a right-angled triangle where the two legs represent the distance traveled east and north, respectively. Let the horizontal leg be $a = 9$ km and the vertical leg be $b = 12$ km. The distance from the starting point to the final position is the hypotenuse $c$ of this right triangle. According to the Pythagorean theorem, $a^{2} + b^{2} = c^{2}$. Substituting the given values, we get $9^{2} + 12^{2} = c^{2}$. Calculating the squares, $81 + 144 = c^{2}$, which simplifies to $225 = c^{2}$. Taking the square root of both sides, $c = \sqrt{225} = 15$. The boat is 15 km from its starting point.