Question

practice 8.2.02 christian lopez problem 4 here are points a and b. point d is the image of point a using point b as the center of dilation and a scale factor of 2. drag the moveable point to plot point d.

Explanation:

Step1: Recall dilation formula

If point $B(x_b,y_b)$ is the center of dilation and point $A(x_a,y_a)$ is being dilated with a scale - factor $k = 2$, the formula for the coordinates of the dilated point $D(x_d,y_d)$ is $(x_d,y_d)=(x_b + k(x_a - x_b),y_b + k(y_a - y_b))$. Geometrically, we can also think of it in terms of vectors. The vector from $B$ to $A$ is $\overrightarrow{BA}=\langle x_a - x_b,y_a - y_b
angle$. When we dilate with a scale - factor of 2 with $B$ as the center, the vector from $B$ to $D$ is $\overrightarrow{BD}=2\overrightarrow{BA}$.

Step2: Locate point D geometrically

Starting from point $B$, we move in the direction of the line segment $BA$. The distance from $B$ to $D$ is twice the distance from $B$ to $A$. So, we extend the line segment $BA$ past point $A$ such that the length of $BD$ is twice the length of $BA$.

Answer:

Locate point $D$ on the line passing through $B$ and $A$ such that the distance from $B$ to $D$ is twice the distance from $B$ to $A$ and $D$ is on the side of $A$ with respect to $B$.