# Illustrative Mathematics Grade 8, Unit 2, Lesson 12: Using Equations for Lines

Learning Targets:

• I can find an equation for a line and use that to decide which points are on that line.

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Illustrative Math

#### Lesson 12: Using Equations for Lines

Let’s write equations for lines.

Illustrative Math Unit 8.2, Lesson 12 (printable worksheets)

#### Lesson 12 Summary

The following diagram shows how to find an equation for a line and use that to decide which points are on that line. #### Lesson 12.1 Missing center

A dilation with scale factor 2 sends A to B. Where is the center of the dilation?

#### Lesson 12.2 Writing Relationships from Two Points

Here is a line.

1. Using what you know about similar triangles, find an equation for the line in the diagram.
2. What is the slope of this line? Does it appear in your equation?
3. Is (9,11) also on the line? How do you know?
4. Is (100,193) also on the line?

#### Are you ready for more?

There are many different ways to write down an equation for a line like the one in the problem. Does (y - 3)/(x - 6) = 2 represent the line? What about (y - 6)/(x - 4) = 5? What about (y + 5)/(x - 1) = 2? Explain your reasoning.

We get select any point on the line for example (7,7)
(7 - 3)/(7 - 6) = 4/1 ≠ 2
(7 - 6)/(7 - 4) = 1/3 ≠ 5
(7 + 5)/(7 - 1) = 12/6 = 2

#### Lesson 12.3 Dilations and Slope Triangles

Here is triangle ABC.

1. Draw the dilation of triangle ABC with center (0,1) and scale factor 2.
2. Draw the dilation of triangle ABC with center (0,1) and scale factor 2.5.
3. Where is C mapped by the dilation with center (0,1) and scale factor s?
4. For which scale factor does the dilation with center (0,1) send C to (9,5.5)? Explain how you know.

#### Lesson 12 Practice Problems

1. Select all the points that are on the line through (0,5) and (2,8).
2. All three points displayed are on the line. Find an equation relating x and y.
3. Here is triangle ABC.
a. Draw the dilation of triangle ABC with center(2, 0)and scale factor 2.
b. Draw the dilation of triangle ABC with center(2, 0)and scale factor 3.
c. Draw the dilation of triangle ABC with center(2, 0)and scale factor 1/2.
d. What are the coordinates of the image of point C when triangle ABC is dilated with center (2,0) and scale factor s?
e. Write an equation for the line containing all possible images of point C.
4. Here are some line segments.
a. Which segment is a dilation of BC using A as the center of dilation and a scale factor of 2/3?
b. Which segment is a dilation of BC using A as the center of dilation and a scale factor of 3/2?
c. Which segment is not a dilation of BC, and how do you know?

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

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