Lesson 12: Using Equations for Lines

Let’s write equations for lines.

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.

  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?

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.

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!

Practice Problems ▶