Lesson 3: Distance on the Coordinate Plane

Session 2. Finding Horizontal and Vertical Side Lengths

Property

To find the length of a side that is perfectly horizontal or vertical, you can either count the grid spaces between the points or find the difference between their coordinates:

• Horizontal lines: Subtract the x-coordinates (the y-coordinates stay the same).
• Vertical lines: Subtract the y-coordinates (the x-coordinates stay the same).

Examples

• The distance between (3, 6) and (10, 6) is a horizontal line. The length is 10 - 3 = 7 units.
• The distance between (4, 2) and (4, 9) is a vertical line. The length is 9 - 2 = 7 units.
• A rectangle has vertices at (1, 1), (6, 1), (6, 5), and (1, 5). Its width (horizontal) is 6 - 1 = 5 units, and its height (vertical) is 5 - 1 = 4 units.

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.