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Lesson 12Using Equations for Lines

Let’s write equations for lines.

Learning Targets:

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

12.1 Missing center

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

Two points labeled A and B with point A below and to the right of point B.

12.2 Writing Relationships from Two Points

Here is a line.

A linear function with points (5,3), (7,7), and (x,y)
  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 \frac{y-3}{x-6}=2 represent the line?  What about \frac{y-6}{x-4}=5 ?  What about \frac{y+5}{x-1}=2 ?  Explain your reasoning.

12.3 Dilations and Slope Triangles

Here is triangle ABC .

Right triangle ABC is on a graph
  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 Summary

We can use what we know about slope to decide if a point lies on a line. Here is a line with a few points labeled.

A line graphed in the x y plane with the origin labeled O. The number 1 through 6 are indicated on each axis. The line begins in quadrant 3, slants upward and to the right passing through the points zero comma one, x comma y, and 2 comma 5.

The slope triangle with vertices (0,1) and (2,5) gives a slope of \frac{5-1}{2-0} =2 . The slope triangle with vertices (0,1) and (x,y) gives a slope of \frac{y-1}{x} . Since these slopes are the same, \frac{y-1}{x} = 2 is an equation for the line. So, if we want to check whether or not the point (11,23) lies on this line, we can check that \frac{23-1}{11} =2 . Since (11,23) is a solution to the equation, it is on the line!

Lesson 12 Practice Problems

  1. Select all the points that are on the line through (0,5) and (2,8) .

    1. (4,11)
    2. (5,10)
    3. (6,14)
    4. (30,50)
    5. (40,60)

  2. All three points displayed are on the line. Find an equation relating x and y .

    A line with points (6,9) and (x,y)

  3. Here is triangle ABC .

    Right triangle ABC is on a graph
    1. Draw the dilation of triangle ABC with center (2,0) and scale factor 2.
    2. Draw the dilation of triangle ABC with center (2,0) and scale factor 3.
    3. Draw the dilation of triangle ABC with center (2,0) and scale factor \frac 1 2 .
    4. What are the coordinates of the image of point C when triangle ABC is dilated with center (2,0) and scale factor s ?
    5. Write an equation for the line containing all possible images of point C .

  4. Here are some line segments.

    Image of a Point A and Lines DE, JG, CB, and HF.
    1. Which segment is a dilation of \overline{BC} using A as the center of dilation and a scale factor of \frac23 ?
    2. Which segment is a dilation of \overline{BC} using A as the center of dilation and a scale factor of \frac32 ?
    3. Which segment is not a dilation of \overline{BC} , and how do you know?