Lesson 86: Calculating the Midpoint and Length of a Segment

New Concept

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

What’s next

Next, you’ll apply this formula to find lengths, classify shapes, and pinpoint the exact center of a line segment using the midpoint formula.

Property

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

Explanation

Think of this formula as the Pythagorean theorem's cool cousin for coordinates! It calculates the direct, straight-line distance between any two points by creating a right triangle from the coordinate differences and finding its hypotenuse. It's a secret map shortcut!

Examples

Find the distance between $(3, -2)$ and $(6, 4)$: $d = \sqrt{(6 - 3)^2 + (4 - (-2))^2} = \sqrt{3^2 + 6^2} = \sqrt{45} = 3\sqrt{5}$.
Find the distance between $(-1, 7)$ and $(4, -5)$: $d = \sqrt{(4 - (-1))^2 + (-5 - 7)^2} = \sqrt{5^2 + (-12)^2} = \sqrt{169} = 13$.

Let's find the exact 'ruler distance' between two points on a map using only their coordinates. This example demonstrates how to use the distance formula, a key idea from this lesson.

Example Problem

Find the distance between $(2, -4)$ and $(7, 8)$.

Property

The midpoint $M$ of the line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is

Explanation

Finding the exact middle of a line is as easy as finding an average, just like the lesson hints! This formula averages the x-values to get the new x-coordinate and averages the y-values for the new y-coordinate, landing you perfectly in the center.

Examples

Find the midpoint of a segment with endpoints $(3, 5)$ and $(7, -2)$: $M = \left(\frac{3 + 7}{2}, \frac{5 + (-2)}{2}\right) = \left(5, \frac{3}{2}\right)$.
Find the midpoint between $(-2, 3)$ and $(4, 7)$: $M = \left(\frac{-2 + 4}{2}, \frac{3 + 7}{2}\right) = \left(\frac{2}{2}, \frac{10}{2}\right) = (1, 5)$.

Finding the dead center of a line is easy—it’s just the average of the endpoints. This example illustrates the midpoint formula, another key idea from this lesson.

Example Problem

Find the midpoint of the line segment with endpoints $(2, 9)$ and $(8, -4)$.