Lesson 5: Using the Pythagorean Theorem

Property

For a triangle with side lengths $a, b,$ and $c$, if the sides satisfy the equation $a^2 + b^2 = c^2$, then the triangle is a right triangle. The right angle is always opposite the longest side, $c$.

Examples

• A triangular garden has sides measuring 8 meters, 15 meters, and 17 meters. Is it a right triangle? Check: $8^2 + 15^2 = 64 + 225 = 289$. The longest side squared is $17^2 = 289$. Yes, it's a right triangle.

• A carpenter builds a frame with sides 5 ft, 12 ft, and a diagonal of 13 ft. Since $5^2 + 12^2 = 25 + 144 = 169$, and $13^2 = 169$, the frame must have a 90-degree corner.

Property

The distance $d$ between points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is

Examples

• To find the distance between $(1, 3)$ and $(5, 6)$, we calculate $d = \sqrt{(5 - 1)^2 + (6 - 3)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$.
• The distance between $(-2, 7)$ and $(3, -5)$ is $d = \sqrt{(3 - (-2))^2 + (-5 - 7)^2} = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
• The distance between $(4, -1)$ and $(-5, -1)$ is $d = \sqrt{(-5 - 4)^2 + (-1 - (-1))^2} = \sqrt{(-9)^2 + 0^2} = \sqrt{81} = 9$.

Explanation

Think of this as the Pythagorean theorem on a coordinate plane. The horizontal change (run) and vertical change (rise) between two points form the legs of a right triangle. The distance formula simply calculates the length of the hypotenuse.