The Distance & Midpoint Formulas

Unit: Parallel & Perpendicular Lines

Today's Goals

Learning Objectives

  • I can calculate the distance between two points on the coordinate plane.
  • I can find the midpoint of a line segment.

The Tools of the Trade

The Distance Formula

Finds the length of a line segment between two points $(x_1, y_1)$ and $(x_2, y_2)$.

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

The Midpoint Formula

Finds the coordinates of the point exactly halfway between two points.

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

I Do: Finding Distance

Let's find the distance between point A(-2, 1) and B(4, 3).

Step 1: Write the Formula

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Step 2: Label Your Points

A($\underset{x_1}{-2}$, $\underset{y_1}{1}$) and B($\underset{x_2}{4}$, $\underset{y_2}{3}$)

Step 3: Substitute and Solve

$d = \sqrt{(4 - (-2))^2 + (3 - 1)^2}$

$d = \sqrt{(6)^2 + (2)^2}$

$d = \sqrt{36 + 4}$

$d = \sqrt{40}$

$d \approx 6.32$ units

I Do: Finding the Midpoint

Let's find the midpoint of the segment with endpoints C(5, -2) and D(-1, 6).

Step 1: Write the Formula

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Step 2: Label Your Points

C($\underset{x_1}{5}$, $\underset{y_1}{-2}$) and D($\underset{x_2}{-1}$, $\underset{y_2}{6}$)

Step 3: Substitute and Solve

$M = \left(\frac{5 + (-1)}{2}, \frac{-2 + 6}{2}\right)$

$M = \left(\frac{4}{2}, \frac{4}{2}\right)$

$M = (2, 2)$

We Do: Let's Practice Together

Problem 1: Distance

Find the distance between E(7, 1) and F(-1, 5).

$d = \sqrt{(\underline{\hspace{1.5cm}} - \underline{\hspace{1.5cm}})^2 + (\underline{\hspace{1.5cm}} - \underline{\hspace{1.5cm}})^2}$

$d = \sqrt{(\underline{\hspace{1.5cm}})^2 + (\underline{\hspace{1.5cm}})^2}$

$d = \sqrt{\underline{\hspace{1.5cm}} + \underline{\hspace{1.5cm}}}$

$d = \sqrt{\underline{\hspace{1.5cm}}}$

Problem 2: Midpoint

Find the midpoint of segment GH where G is (0, -4) and H is (3, 2).

$M = \left(\frac{\underline{\hspace{1.5cm}} + \underline{\hspace{1.5cm}}}{2}, \frac{\underline{\hspace{1.5cm}} + \underline{\hspace{1.5cm}}}{2}\right)$

$M = \left(\frac{\underline{\hspace{1.5cm}}}{2}, \frac{\underline{\hspace{1.5cm}}}{2}\right)$

$M = (\underline{\hspace{1.5cm}}, \underline{\hspace{1.5cm}})$

You Do: Your Turn!

Solve the following problems on your own. Show your work.

Problem 1: Distance

What is the length of the segment connecting J(6, 8) and K(-3, 8)?

Problem 2: Midpoint

Find the midpoint of the segment with endpoints L(10, 3) and N(-4, -5).

Independent Practice

Choose a level and complete the problems. Show all your work.

RED LEVEL

  1. Find the distance between (1, 2) and (4, 6).
  2. Find the midpoint of a segment with endpoints (2, 5) and (8, 1).

YELLOW LEVEL

  1. Find the distance between (-3, 5) and (4, -1). (Simplify the radical if possible).
  2. Find the midpoint of a segment with endpoints (-7, 2) and (3, -8).

GREEN LEVEL

  1. A segment has an endpoint at (2, 8) and a midpoint at (5, 2). Find the other endpoint.
  2. Find the perimeter of a triangle with vertices A(1,5), B(5,2), and C(1,2).

Exit Ticket

Show me what you learned! Answer the questions below.

1. Find the distance between P(-5, -2) and Q(3, 4).

2. Find the midpoint of the segment connecting R(9, -1) and S(-1, 5).

3. In your own words, when would you use the distance formula vs. the midpoint formula?