Exercise 14 Page 677

To draw the image of a figure under a dilation with center $S$ and the scale factor $k,$ we need to follow a few steps.

1. Draw rays from $S$ through each vertex.
2. Choose one vertex, which we will arbitrarily call $A.$ Locate $A^{\prime }$ on $\overrightarrow{SA}$ such that $S{A}^{\prime }=kSA.$
3. Repeat Step $2$ for the remaining vertices. Then connect the new vertices to create the image.

Let's draw the dilation of the given figure!

Step $1$

Notice that $S$ is one of the vertices of the original figure. Therefore, the image and the preimage are the same point $S.$ We will begin by drawing the rays from $S$ through each vertex. Let's also name the remaining vertices $A,$ $B,$ and $C.$

Step $2$

The given scale factor is $k=\frac{7}{4}.$ Let's locate $A^{\prime }$ on $\overrightarrow{SA}$ such that $S{A}^{\prime }=\frac{7}{4}SA.$ To do this, we measure $\overline{SA}$ first. Then, after multiplying $SA$ by $\frac{7}{4},$ we get the length of $\overline{S{A}^{\prime }}.$ We locate point $A^{\prime }$ by measuring this length from $S$ along $\overrightarrow{SA}.$

Step $3$

We can locate $B^{\prime }$ on $\overrightarrow{SB}$ and $C^{\prime }$ on $\overrightarrow{SC}$ in the same way we located $A^{\prime }$ in Step $2.$

Finally, we can connect the new vertices and get the image of the original figure.