Lesson 6.2 – The Distance Formula

Concept:

EQ:

Vocabulary:

How do we find distances in the coordinate plane? (G.GPE.7)

Distance in the Coordinate Plane

Pythagorean TheoremDistance formulaSquare rootSquared

How far apart are the points (-4, 1) and (2, 4)?

(2, 4)

(-4, -1)

Activator

Let’s Review

The Pythagorean Theorem

a

bc

a2 + b2 = c2

Core Lesson

If you are given two points on a plane, you can draw a right triangle with the

points as vertices.

(2, 4)

(-4, -1)

Core Lesson

Find the third vertex using your two points. Then find the vertical and

horizontal distances.

(2, 4)

(-4, -1) (2, -1)

5

6

Core Lesson

Once you have your horizontal and vertical distances, you can apply the

Pythagorean Theorem.

b=5

a=6

ca2 + b2 = c2

62 + 52 = c2

36 + 25 = c2

c2 = 61c =

Core Lesson

If you use variables in place of real values, you can derive a formula to calculate the distance

between any two points.

(x1, y1)

(x2, y2) (x1, y2)

Core Lesson

(x1, y1)

(x2, y2) (x1, y2)

To find the horizontal and vertical distances, find the differences between the x and y

values respectively.

a

bcThis is called the Distance

Formula.

Core Lesson

Use the distance formula to find the distance between the two given points.

(2, 4)

(-4, -1)

x1 = 2, y1 = 4

x2 = -4, y2 = -1

Core Lesson

The Distance Formula is a formula that allows you to find the distance between two

points on a coordinate plane.

Core Lesson

Steps to using the Distance Formula:

1. Label the points ( and .2. Write the Distance formula.3. Substitute your points into the Distance

Formula.4. Evaluate using your calculator.

Core Lesson Guided Practice - Example 1

Find the distance between the two points (2, 3) and (4, 5)*Round the result to the nearest hundredth if necessary.

Step 1: Label the points and . Step 3: Substitute your points into the distance formula.

Step 2: Write the Distance Formula.

Step 4: Evaluate using your calculator.

Core Lesson

Core Lesson Guided Practice - Example 2

Find the distance between the two points (0, 4) and (-3, 0)*Round the result to the nearest hundredth if necessary.

Step 1: Label the points and . Step 3: Substitute your points into the distance formula.

Step 2: Write the Distance Formula.

Step 4: Evaluate using your calculator.

Core Lesson You Try - 1

Find the distance between the two points (-4, 2) and (1, 4)*Round the result to the nearest hundredth if necessary.

Step 1: Label the points and . Step 3: Substitute your points into the distance formula.

Step 2: Write the Distance Formula.

Step 4: Evaluate using your calculator.

Core Lesson

When applying the distance formula, directions are often used to describe the location of a point.

Applying the Distance Formula

Core Lesson Guided Practice - Example 3

From your home, you ride your bicycle 5 miles north, then 12 miles east. How far are you from your home?  

Prior to Step 1: Write your points based on the direction.

5 miles north: x or y = ______ 12 miles east: x or y = ______ ( _____, _____ )

(Circle one) (Circle one)

Home: ( _____, _____ )

Core Lesson Guided Practice - Example 3From your home, you ride your bicycle 5

miles north, then 12 miles east. How far are you from your home?Step 1: Label the points and . Step 3: Substitute your points

into the distance formula.

Step 2: Write the Distance Formula.

Step 4: Evaluate using your calculator.

You are ________ miles from your home.

Core Lesson Guided Practice - Example 4

Plane 1 is located six miles east and two miles south of an airport. Plane 2 is located one mile east and 10 miles north of the same airport. Find the distance between the planes. 

Prior to Step 1: Write your points based on the direction.

Plane 1: 6 miles east: x or y = ______ ; 2 miles south: x or y = ______ ( _____, _____ )

Plane 2: 1 mile east: x or y = ______ ; 10 miles north: x or y = ______ ( _____, _____ )

Circle one

Circle one

Circle one

Circle one

Core Lesson Guided Practice - Example 4Plane 1 is located six miles east and two miles south of an

airport. Plane 2 is located one mile east and 10 miles north of the same airport. Find the distance between the planes. 

Step 1: Label the points and . Step 3: Substitute your points into the distance formula.

Step 2: Write the Distance Formula.

Step 4: Evaluate using your calculator.

The planes are ________ miles apart.

Core Lesson You Try 2At a state park, Fred and Ben’s campsite is located three miles west

and six miles north of the ranger station. Lizzie and Roxy’s campsite is located four miles east and two miles south of the ranger station. Find the distance between the campsites. 

Prior to Step 1: Write your points based on the direction.

Fred and Ben’s Campsite:

3 miles west: x or y = ______ ; 6 miles north: x or y = ______ ( _____, _____ )

Lizzie and Roxy’s Campsite:

4 miles east: x or y = ______ ; 2 miles south: x or y = ______ ( _____, _____ )

Circle one

Circle one

Circle one

Circle one

Core Lesson You Try 2At a state park, Fred and Ben’s campsite is located three miles west and six miles north of the ranger station. Lizzie and Roxy’s campsite is located four miles east and two miles south of the ranger station. Find the distance between the campsites. Step 1: Label the points and . Step 3: Substitute your points

into the distance formula.

Step 2: Write the Distance Formula.

Step 4: Evaluate using your calculator.

The campsites are ________ miles apart.

On a sheet of paper, write down three things you learned today. Out of those three, write which one is most important and why.

The Important Thing