1. Activating Prior Knowledge
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Activating Prior Knowledge
Find the missing side of the following right triangles:
a = 6, b = a = 5, c = 9
Decide if the following side lengths would form a right triangle.
3. 6, 8, 10
4. 4, 8, 9
Tie to LO

2. Learning Objective
Today, we will determine the distance between two points on a coordinate plane using the Pythagorean Theorem.
CFU

3. The distance between points 𝐴, 𝐵 is 6 units.
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Concept Development
Example 1
What is the distance between the two points 𝑨, 𝑩 on the coordinate plane?
The distance between points 𝐴, 𝐵 is 6 units.
CFU

4. The distance between points 𝐴, 𝐵 is 2 units.
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Concept Development
Example 1 (continued)
What is the distance between the two points 𝑨, 𝑩 on the coordinate plane?
The distance between points 𝐴, 𝐵 is 2 units.
CFU

5. CFU Concept Development Lesson 17: Distance on the Coordinate Plane
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Concept Development
Example 1 (continued)
We cannot simply count units between the points because the line that connects 𝐴 to 𝐵 is not horizontal or vertical. What have we done recently that allowed us to find the length of an unknown segment?
The Pythagorean Theorem allows us to determine the length of an unknown side of a right triangle.
CFU

6. The distance between points 𝐴, 𝐵 is 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 6.3 units.
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Concept Development
Example 1 (continued)
Draw a vertical line through 𝐵 and a horizontal line through 𝐴. Or, draw a vertical line through 𝐴 and a horizontal line through 𝐵.
Let’s mark the point of intersection of the horizontal and vertical lines we drew as point 𝐶. What is the length of 𝐴𝐶 ? 𝐵𝐶 ?
The length of 𝐴𝐶 =6 units, and the length of 𝐵𝐶 =2 units.
The distance between points 𝐴, 𝐵 is 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 6.3 units.
CFU

7. CFU Concept Development Lesson 17: Distance on the Coordinate Plane
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Concept Development
Example 2 Given two points 𝑨, 𝑩 on the coordinate plane, determine the distance between them. First, make an estimate; then, try to find a more precise answer. Round your answer to the tenths place.
CFU

8. CFU Concept Development Lesson 17: Distance on the Coordinate Plane
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Concept Development
Example 2 (continued) We know that we need a right triangle. How can we draw one?
Draw a vertical line through 𝐵 and a horizontal line through 𝐴. Or draw a vertical line through 𝐴 and a horizontal line through 𝐵.
CFU

9. CFU Concept Development Lesson 17: Distance on the Coordinate Plane
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Concept Development
Example 2 (continued) The length 𝐴𝐶 =3 units, and the length 𝐵𝐶 =3 units. Let 𝑐 be |𝐴𝐵|.
The distance between points 𝐴 and 𝐵 is approximately 4.2 units.
CFU

10. Skilled Development/Guided Practice
M7:LSN16
The Converse of the Pythagorean Theorem
Skilled Development/Guided Practice
Exercises 1-4
For each of the Exercises 1–4, determine the distance between points 𝑨 and 𝑩 on the coordinate plane. Round your answer to the tenths place.
10 minutes
End
CFU

11. Skilled Development/Guided Practice Exercises 1-4 (answer check)
M7:LSN16
The Converse of the Pythagorean Theorem
Skilled Development/Guided Practice
Exercises 1-4 (answer check)
The distance between points 𝑨 and 𝑩 is about 𝟕.𝟖 units.
The distance between points 𝑨 and 𝑩 is about 𝟏𝟑.𝟔 units.
The distance between points 𝑨 and 𝑩 is about 𝟓.𝟖 units.
The distance between points 𝑨 and 𝑩 is about 𝟔.𝟒 units.
CFU

12. CFU Concept Development Lesson 17: Distance on the Coordinate Plane
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Concept Development
Example 3
Is the triangle formed by the points 𝑨, 𝑩, 𝑪 a right triangle?
How can we verify if a triangle is a right triangle?
Use the converse of the Pythagorean Theorem.
What information do we need about the triangle in order to use the converse of the Pythagorean Theorem, and how would we use it?
We need to know the lengths of all three sides; then, we can check to see if the side lengths satisfy the Pythagorean Theorem.
CFU

13. CFU Concept Development Lesson 17: Distance on the Coordinate Plane
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Concept Development
Example 3 (continued)
Clearly, the length of 𝐴𝐵 =10 units. How can we determine 𝐴𝐶 ?
To find 𝐴𝐶 , follow the same steps used in the previous problem. Draw horizontal and vertical lines to form a right triangle, and use the Pythagorean Theorem to determine the length.
CFU

14. CFU Concept Development Lesson 17: Distance on the Coordinate Plane
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Concept Development
Example 3 (continued)
Determine 𝐴𝐶 . Leave your answer in square root form unless it is a perfect square.
Let 𝑐 represent |𝐴𝐶|.
= 𝑐 = 𝑐 2 10= 𝑐 =𝑐
CFU

15. CFU Concept Development Lesson 17: Distance on the Coordinate Plane
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Concept Development
Example 3 (continued)
Now, determine 𝐵𝐶 . Again, leave your answer in square root form unless it is a perfect square.
Let 𝑐 represent |𝐵𝐶|.
= 𝑐 = 𝑐 2 90= 𝑐 =𝑐
CFU

16. CFU Concept Development Lesson 17: Distance on the Coordinate Plane
M7:LSN17
Lesson 17: Distance on the Coordinate Plane
Concept Development
Example 3 (continued)
The lengths of the three sides of the triangle are 10 units, units, and units. Which number represents the hypotenuse of the triangle? Explain.
The side 𝐴𝐵 must be the hypotenuse because it is the longest side. When estimating the lengths of the other two sides, I know that is between 3 and 4, and is between 9 and 10. Therefore, the side that is 10 units in length is the hypotenuse.
Use the lengths 10, 10 , and to determine if the triangle is a right triangle.
= = =100
Therefore, the points 𝐴, 𝐵, 𝐶 form a right triangle.
CFU

17. M7:LSN17
Lesson 17: Distance on the Coordinate Plane
LESSON SUMMARY
To determine the distance between two points on the coordinate plane, begin by connecting the two points. Then draw a vertical line through one of the points and a horizontal line through the other point. The intersection of the vertical and horizontal lines forms a right triangle to which the Pythagorean Theorem can be applied.
To verify if a triangle is a right triangle, use the converse of the Pythagorean Theorem.
CFU

18. Closure Homework – Module 7 page 88-90 Problem Set #1-5 CFU
What did we learn today?
How do you find the distance between two vertical or horizontal points on a coordinate plane?
3. How do you find the distance between two points on a coordinate plan that are not vertical or horizontal?
Homework – Module 7 page Problem Set #1-5
CFU

19. For each of the Problems 1–4, determine the distance between points A and B on the coordinate plane. Round your answer to the tenths place.
Homework

20. For each of the Problems 1–4, determine the distance between points A and B on the coordinate plane. Round your answer to the tenths place.
Homework

21. For each of the Problems 1–4, determine the distance between points A and B on the coordinate plane. Round your answer to the tenths place.
Homework

22. For each of the Problems 1–4, determine the distance between points A and B on the coordinate plane. Round your answer to the tenths place.
Homework

23. Homework
Is the triangle formed by points 𝐴, 𝐵, 𝐶 a right triangle?