Question

1. a flag is represented by the coordinates a(3, 9), b(15, 9), c(15, 3), and d(3, 3). suppose you were to dilate the figure by a scale factor of ⅓ using the origin as the center of dilation.

a. what are the coordinates of the dilated figure?
b. compare and contrast the corresponding angles and corresponding side - lengths of the original figure and the dilated figure.
c. is the image congruent to the pre - image? why or why not?
d. how does the perimeter of the original figure compare to the perimeter of the dilated figure?

Explanation:

Step1: Recall dilation formula

For a point $(x,y)$ dilated by a scale - factor $k$ with the origin as the center of dilation, the new coordinates $(x',y')$ are given by $(kx,ky)$. Here the scale factor $k = \frac{1}{3}$.

Step2: Calculate new coordinates

For point $A(3,9)$:
$x'_A=\frac{1}{3}\times3 = 1$, $y'_A=\frac{1}{3}\times9 = 3$.
For point $B(15,9)$:
$x'_B=\frac{1}{3}\times15 = 5$, $y'_B=\frac{1}{3}\times9 = 3$.
For point $C(15,3)$:
$x'_C=\frac{1}{3}\times15 = 5$, $y'_C=\frac{1}{3}\times3 = 1$.
For point $D(3,3)$:
$x'_D=\frac{1}{3}\times3 = 1$, $y'_D=\frac{1}{3}\times3 = 1$.
So the coordinates of the dilated figure are $A'(1,3)$, $B'(5,3)$, $C'(5,1)$, $D'(1,1)$.

Step3: Analyze angle and side - length relationships

a. Angles in a dilated figure and the original figure are congruent. This is because dilation is a similarity transformation, and similar figures have the same angle measures.
b. The side - lengths of the dilated figure are related to the original figure by the scale factor. If the scale factor is $k=\frac{1}{3}$, the side - lengths of the dilated figure are $\frac{1}{3}$ of the side - lengths of the original figure.
c. The image is not congruent to the pre - image because the side - lengths have changed. Congruent figures have equal side - lengths, and here the side - lengths of the dilated figure are $\frac{1}{3}$ of the original side - lengths.
d. If the perimeter of the original figure is $P$ and the scale factor is $k$, the perimeter of the dilated figure $P'$ is related to $P$ by $P'=kP$. Since $k = \frac{1}{3}$, the perimeter of the dilated figure is $\frac{1}{3}$ of the perimeter of the original figure.

Answer:

a. The angles of the dilated figure and the original figure are congruent.
b. The side - lengths of the dilated figure are $\frac{1}{3}$ of the side - lengths of the original figure.
c. The image is not congruent to the pre - image because the side - lengths have changed.
d. The perimeter of the dilated figure is $\frac{1}{3}$ of the perimeter of the original figure.