Question

△def has vertices d(9,3), e(18,6), and f(24,3). △def has vertices d(3,1), e(6,2), and f(8,1). (a) complete the following. the line through e and f does not pass through the center of dilation. the line through d and e passes through the center of dilation. (b) find each slope below. slope of (overrightarrow{ef}) = slope of (overrightarrow{ef}) = slope of (overrightarrow{de}) = slope of (overrightarrow{de}) =

Explanation:

Step1: Recall slope - formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate slope of $\overrightarrow{EF}$

For points $E(18,6)$ and $F(24,3)$, using the slope formula:
$m_{EF}=\frac{3 - 6}{24 - 18}=\frac{- 3}{6}=-\frac{1}{2}$

Step3: Calculate slope of $\overrightarrow{E'F'}$

For points $E'(6,2)$ and $F'(8,1)$, using the slope formula:
$m_{E'F'}=\frac{1 - 2}{8 - 6}=\frac{-1}{2}=-\frac{1}{2}$

Step4: Calculate slope of $\overrightarrow{DE}$

For points $D(9,3)$ and $E(18,6)$, using the slope formula:
$m_{DE}=\frac{6 - 3}{18 - 9}=\frac{3}{9}=\frac{1}{3}$

Step5: Calculate slope of $\overrightarrow{D'E'}$

For points $D'(3,1)$ and $E'(6,2)$, using the slope formula:
$m_{D'E'}=\frac{2 - 1}{6 - 3}=\frac{1}{3}$

Answer:

slope of $\overrightarrow{EF}$ = slope of $\overrightarrow{E'F'}=-\frac{1}{2}$
slope of $\overrightarrow{DE}$ = slope of $\overrightarrow{D'E'}=\frac{1}{3}$