Lesson 41: Using the Pythagorean Theorem and the Distance Formula (Exploration: Visualizing the Pythagorean Theorem)

New Concept

The distance $d$ between any two points with coordinates $(x_1, y_1)$ and $(x_2, y_2)$ is

What’s next

Next, you’ll apply this formula to find lengths, verify triangles, and solve problems on a coordinate grid.

Property

If a triangle is a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. For legs $a$ and $b$, and hypotenuse $c$, $a^2 + b^2 = c^2$.

A right triangle has a leg of 9 cm and a hypotenuse of 15 cm. Find the other leg, $b$: $9^2 + b^2 = 15^2 \rightarrow 81 + b^2 = 225 \rightarrow b^2 = 144 \rightarrow b = 12$ cm.
A right triangle has legs of 7 in and 24 in. Find the hypotenuse, $c$: $7^2 + 24^2 = c^2 \rightarrow 49 + 576 = c^2 \rightarrow 625 = c^2 \rightarrow c = 25$ in.

Think of this as a magic recipe for right triangles! If you know the lengths of the two shorter sides (the legs), you can find the length of the longest side, the hypotenuse. Just square the legs, add them up, and the result is the square of the hypotenuse. It's a trusty shortcut for finding any missing side!

Property

If the sum of the squares of the lengths of the two shorter sides of a triangle equals the square of the length of the longest side, then the triangle is a right triangle.

Do side lengths 8, 15, and 17 form a right triangle? Check: $8^2 + 15^2 \stackrel{?}{=} 17^2 \rightarrow 64 + 225 \stackrel{?}{=} 289 \rightarrow 289 = 289$. Yes, it is a right triangle.
Do side lengths 10, 20, and 23 form a right triangle? Check: $10^2 + 20^2 \stackrel{?}{=} 23^2 \rightarrow 100 + 400 \stackrel{?}{=} 529 \rightarrow 500 \neq 529$. Nope, not a right triangle.

This is the detective version of the theorem! It lets you test if any triangle is secretly a right triangle. Just take the three side lengths, plug them into $a^2 + b^2 = c^2$, making sure 'c' is the longest side. If the equation is true, you've found a right triangle. If not, the triangle is just an imposter.

Property

The distance $d$ between any two points with coordinates $(x_1, y_1)$ and $(x_2, y_2)$ is

Find the distance between $(2, 3)$ and $(10, 9)$: $d = \sqrt{(10 - 2)^2 + (9 - 3)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$.
Find the distance between $(-1, 7)$ and $(4, -5)$: $d = \sqrt{(4 - (-1))^2 + (-5 - 7)^2} = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.

This formula looks fancy, but it's just the Pythagorean theorem dressed up for a coordinate plane party. It finds the direct, straight-line distance between any two points. Imagine drawing a right triangle connecting the points; the horizontal and vertical distances are the legs. The formula just calculates the hypotenuse, which is the distance you actually wanted to find.