Find Distances on the Coordinate Plane

Key Concepts

• Find vertical distance.
• Find horizontal distance.
• Solve problems using distance.

Introduction: 

Finding the distance between two points on a coordinate plane: 

When two points are in the same quadrant, we subtract to find the difference of their absolute values to find the distance between them. 

When two points are in different quadrants, we add to find the difference of their absolute values to find the distance between them. 

• If the x-coordinates are the same, we use the y-coordinates to find the distance. 
• If the y-coordinates are the same, we use the x-coordinates to find the distance. 

For example, if the coordinates are (3, 5) and (3, -2), use y-coordinates. 

If the coordinates are (-1, 4) and (6, 4), use x-coordinates. 

2.5.1 Find vertical distance 

Example 1: 

Find the distance between (-4, –1) and (-4, –6).  

Solution: 

Use absolute value. Look at the y-coordinates. Since the points are in the same quadrant, subtract the absolute values.  

|-6| – |-1| = 6 – 1 = 5.  

Example 2: 

Find the distance from the Ferris wheel to roller coaster 3. 

Solution: 

Find the coordinates of the Ferris wheel and the roller coaster 3. 

• The coordinates of the Ferris wheel are (-6, 2). 
• The coordinates of the roller coaster 3 are (-6, -8). 

The absolute value of the y-coordinates tells you the distance between each point and the x-axis. 

The distance from Ferris wheel to roller coaster 3 is 

|2|+|-8| = 2 + 8 = 10 miles. 

2.5.2 Find horizontal distance 

Example 3: 

Find the distance between (-3, 5) and (4, 5).  

Solution: 

Use absolute value. Look at the x-coordinates. Since the points are in the different quadrants, add the absolute values.  

|-3| + |4| = 3 + 4 = 7.  

Example 4: 

Find the distance from roller coaster 1 to the swings. 

Solution: 

Find the coordinates of the roller coaster 1 and the swings. 

• The coordinates of the roller coaster 1 are (-6, 7). 
• The coordinates of the roller swings are (1, 7). 

The distance from roller coaster 1 to the swings is 

|-6|+|1| = 6 + 1 = 7 miles. 

2.5.3 Solve problems using distance 

Example 5: 

The graph shows the locations of point G and point H. Point J is graphed at (n, -3). The distance from point H to point J is equal to the distance from point H to point G. 

1. What is the distance from point H to point J? 

2. What is the value of n? 

Solution: 

Step 1: 

Find the distance from point H to point J. 

The x-coordinates are the same. 

|-6|-|-3| = 6 – 3 = 3 units.  

Step 2: 

The distance from point H to point G is 3 units. 

n – 0 =3 

n = 3 

Exercise:

1. To find distances between points on a coordinate plane, we need to use _______________.
2. We need to ___________ absolute values to find distance between two points using their coordinates if they lie in different quadrants.
3. We need to _____________ absolute values to find distance between two points using their coordinates if they lie in the same quadrant.
4. Find the distance between the points (5, —6) and (2, —6)
5. Find the distance between each pair of points (-6, —4.7) and (-6, 4.1)
6. Find the distance between the points (- 2 , 1 ) and (-1 , 1 )
7. Find the distance between the points (-7, -4) and (-7, 9)
8. Find the distance from the Roller Coaster 1 to Ferris Wheel.

9.Find the distance between the Water Park and the Coulter’s Home.

10. Find the distance between the school and the store.

Concept Map 

What have we learned:

• Find vertical distance between two points on a coordinate plan.
• Find horizontal distance between two points on a coordinate plan.
• Solve problems using distance on a coordinate plan.