lesson 25 introduction m.8.23 distance in the coordinate plane€¦ · introduction 212...

Introduction

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Use What You Know

Lesson 25 Distance in the Coordinate Plane

Lesson 25Distance in the Coordinate Plane

You have solved problems with two- and three-dimensional figures using the Pythagorean Theorem. Take a look at this problem.

What is the distance between (2, 3) and (5, 7) in the coordinate plane?

RP a

y

x(2, 3)

Q (5, 7)

Use math you already know to solve the problem.

a. Write the equation for the Pythagorean Theorem for your reference.

b. How do you know triangle PQR is a right triangle?

c. Label the legs of triangle PQR a and b and label the hypotenuse c.

d. What is the distance from point P to point R?

e. What is the distance from point Q to point R?

f. Write a numerical expression for the length of the hypotenuse, c.

g. Explain how you can find the distance between points P and Q.

M.8.23

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Find Out More


The Pythagorean Theorem gives you the tools to find the distance between any two given points in the coordinate plane.

y

x?

S

T

If you know the coordinates of any two points you can find the distance between them. Draw a vertical line through one point and a horizontal line through the other point to create a right triangle.

Find the distance from each point to the vertex of the right angle. Then use those distances as the lengths of the legs. Use the equation for the Pythagorean Theorem to find an expression for the length of the hypotenuse. That is the distance between the two given points.

Reflect1 Describe how finding the distance between two points in the coordinate plane is similar

to finding the length of the hypotenuse of a triangle.

T

S

c

y

xb

a

Pythagorean triples can come in handy in the coordinate plane, too.

Modeled and Guided Instruction

Learn About


Lesson 25


Finding the Distance Between Two Points

Read the problem below. Then explore different ways to find the distance between two points in the coordinate plane.

What is the distance between the points (25, 23) and (22, 4)?

y

x

P(25, 23)

Q(22, 4 )

Picture It You can sketch a right triangle.

Draw a vertical line segment from one point and a horizontal line segment from the other. PQR is a right triangle. PR and QR are legs. PQ is the hypotenuse.

Model It You can use equations to represent the lengths of the legs.

a 5 |22 2 (25)|

b 5 |4 2 (23)|

c2 5 a2 1 b2

a

c b

y

x

Q(22, 4)

P(25, 23) R(22, 23)

©Curriculum Associates, LLC Copying is not permitted. 215Lesson 25 Distance in the Coordinate Plane

Connect It Now you will use equations to solve the problem.

2 Explain how you know the equation PR2 1 QR2 5 PQ2 is true for triangle PQR on the

previous page.

3 What is the length of a?

4 What is the length of b?

5 What is the length of c?

6 What is the distance between (25, 23) and (22, 4)?

7 Explain how to find the distance between any two points in the coordinate plane that do

not lie on the same horizontal or vertical line.

Try It Use what you just learned to solve these problems. Show your work on a separate sheet of paper.

8 Find the distance between points J and K.

9 Find the distance between points K and L.

10 Classify triangle JKL and justify your answer.

y

x

L (7, 23)J (25,23)

K (1, 5)

Guided Practice

Practice


Lesson 25



Study the example below. Then solve problems 11–13.

Example

Classify the triangle formed by points (21, 2), (1, 21) and (22, 21) as scalene, isosceles, or equilateral. Justify your answer.

Look at how you could show your work.

y

x

A

B

C

Solution

11 Find the distance between the points (3, 22) and (25, 4).

Show your work.

Solution

Pair/ShareWhy did the student not find exact values for the side lengths?

Pair/ShareWhat other ways can you find the solution?

I guess I need to use the Pythagorean Theorem two times.

Where’s the right triangle?

The triangle is scalene; all three sides have different lengths.

AB 5 Ï······· 12 1 32 5 Ï··· 10

BC 5 Ï······· 22 1 32 5 Ï··· 13

AC 5 3

Ï··· 10 Þ Ï··· 13 Þ 3

y

x

(3,22)

(25, 4)


12 Draw a line segment from the origin (0, 0) with length Ï··· 10 .

Show your work.

y

x

Solution

13 Which expression can be used to find the distance between points (21, 21) and (19, 20)?

A AB 5 Ï······· 22 1 42 5 Ï··· 20

B d 5 Ï················ (19 21)2 1 (20 21)2

C d 5 Ï·················· (2 121)2 1 (20 219)2

D d 5 Ï·················· (19 1 1)2 1 (20 1 1)2

Adam chose B as the correct answer. How did he get that answer?

Pair/ShareHow many different solutions to this problem can you find?

Pair/ShareIf you check your answer with a calculator does it seem reasonable?

How does this relate to a right triangle?

Hmm. I need a right triangle with hypotenuse Ï··· 10 .

Independent Practice

Practice


Lesson 25



Solve the problems.

1 Segment AB begins at A(2, 1) and ends at point B(6, 4). The segment is dilated by a factor of 2 with the center of dilation at the origin to form segment A9B9. Segment A9B9 is then translated 4 units left and 2 units down to form segment A99B99. Choose True or False for each statement.

a. Segment A99B99 is congruent to segment AB. True False

b. The coordinates of A99 are those of the origin. True False

c. The distance between A and B is one half the distance between A9 and B9. True False

d. Segment A99B99 has the same length as segment A9B9. True False

e. The length of A99B99 would be different if the translation had occurred before the dilation. True False

2 Decide which of the following points, A through F, belong in each of the three categories in the table. Then write the letter of the point in the proper column.

A (27, 25) B (14, 24) C (6, 8) D (2, 8) E (9, 9) F (6, 28)

A point greater than 10 units from the origin

A point exactly 10 units from the origin

A point less than 10 units from the origin

3 Find the perimeter of parallelogram ABCD with vertices at (23, 3), (3, 3), (0, 21), and (26, 21).

Show your work.

Answer

A B

D C

y

x

Self Check


Go back and see what you can check off on the Self Check on page 159.

4 Draw the reflection of triangle ABC in the y-axis. Then show that the corresponding sides of the two triangles are congruent.

Show your work.

5 Segment AB is one side of a square. Find the coordinates of the other two vertices of the square and draw the square. Explain your reasoning. More than one answer is possible.

y

x

A

B (2, 1)

(22, 3)

y

x

A (2, 2) B (5, 2)

C (4, 22)

lesson 25 introduction m.8.23 distance in the coordinate plane€¦ · introduction 212...

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