## 12.1: Missing center

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

Let’s write equations for lines.

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

Here is a line.

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

Here is triangle $ABC$.

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

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.

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!