Question

1 dilating lines
here is a diagram with points a, b, c, d, and e.
1 dilate point a using center c and a scale factor of $\frac{3}{4}$.
2 dilate point b using center c and a scale factor of $\frac{1}{3}$.
3 dilate point d using center c and a scale factor of $\frac{3}{2}$.
4 dilate line ce using center c and a scale factor of 2.
5 what happens when the center of dilation is on a line and then you dilate the line?

Explanation:

Step1: Recall dilation formula for points

If we have a center of dilation $C(x_c,y_c)$ and a point $P(x_p,y_p)$ and a scale - factor $k$, the coordinates of the dilated point $P'(x_{p'},y_{p'})$ are given by $x_{p'}=x_c + k(x_p - x_c)$ and $y_{p'}=y_c + k(y_p - y_c)$. In a non - coordinate geometric sense, if we consider the vector from $C$ to $P$, we multiply the length of this vector by $k$. For a line dilation with center $C$ on the line, we consider the distance of points on the line from $C$.

Step2: Dilate point A

Let's assume we are working in a non - coordinate geometric space. The distance from $C$ to $A$ is some length $d_{CA}$. After dilation with a scale factor $k = \frac{3}{4}$, the new point $A'$ lies on the line $CA$ and the distance from $C$ to $A'$ is $d_{CA'}=\frac{3}{4}d_{CA}$.

Step3: Dilate point B

The scale factor for dilating point $B$ is $k=\frac{1}{3}$. The new point $B'$ lies on the line $CB$ and the distance from $C$ to $B'$ is $d_{CB'}=\frac{1}{3}d_{CB}$.

Step4: Dilate point D

The scale factor for dilating point $D$ is $k = \frac{3}{2}$. The new point $D'$ lies on the line $CD$ and the distance from $C$ to $D'$ is $d_{CD'}=\frac{3}{2}d_{CD}$.

Step5: Dilate line CE

When we dilate line $CE$ with center $C$ and scale factor $k = 2$, for any point $Q$ on $CE$, the dilated point $Q'$ lies on the ray from $C$ through $Q$ and the distance from $C$ to $Q'$ is $2$ times the distance from $C$ to $Q$. The line $CE$ after dilation is still a line passing through $C$, and all points on the new line are at a distance from $C$ that is $2$ times the distance of the corresponding original points from $C$.

Step6: Analyze line dilation with center on the line

When the center of dilation $C$ is on the line (like line $CE$), the dilated line is the same line. This is because for any point $X$ on the line, the dilated point $X'$ with center $C$ on the line lies on the same line. The direction of the line does not change, and since $C$ is fixed (because the distance from $C$ to $C$ is $0$ and $k\times0 = 0$), and all other points on the line are just stretched or compressed along the line with respect to $C$, the line remains in the same position.

Answer:

1. The dilated point $A'$ lies on the line $CA$ and the distance from $C$ to $A'$ is $\frac{3}{4}$ of the distance from $C$ to $A$.
2. The dilated point $B'$ lies on the line $CB$ and the distance from $C$ to $B'$ is $\frac{1}{3}$ of the distance from $C$ to $B$.
3. The dilated point $D'$ lies on the line $CD$ and the distance from $C$ to $D'$ is $\frac{3}{2}$ of the distance from $C$ to $D$.
4. The dilated line $CE$ is a line passing through $C$ where the distance of each point on the new line from $C$ is $2$ times the distance of the corresponding original point from $C$.
5. When the center of dilation is on a line, the dilated line is the same line.