Question

1 2 assignment
write
in your own words, explain how to dilate a figure on the coordinate plane using repeated translations. use examples with scale factors between 0 and 1 and greater than 1 to illustrate your explanation.
remember
if the dilation of a figure is centered at the origin, you can multiply the coordinates of the points of the original figure by the scale - factor to determine the coordinates of the new figure.
practice

1. graph △xyz with the coordinates x(3, 18), y(18, 18), and z(18, 9).

a. reduce △xyz on the coordinate plane using the origin as the center of dilation and a scale factor of $\frac{1}{3}$ to form △xyz.
b. what are the coordinates of points x, y, and z?
x(1, 6), y(6, 6), z(6, 3)
c. what is the area of the pre - image and the image?
d. what is the relationship between the two areas?
e. if the perimeter of the pre - image is 41.49 units, what is the perimeter of the image? explain your reasoning.

Explanation:

Step1: Find coordinates of dilated points

When dilating a point $(x,y)$ with the origin as the center of dilation and a scale - factor $k$, the new coordinates $(x',y')$ are given by $(kx,ky)$. Here $k = \frac{1}{3}$. For point $X(3,18)$, $X'=(3\times\frac{1}{3},18\times\frac{1}{3})=(1,6)$; for point $Y(18,18)$, $Y'=(18\times\frac{1}{3},18\times\frac{1}{3})=(6,6)$; for point $Z(18,9)$, $Z'=(18\times\frac{1}{3},9\times\frac{1}{3})=(6,3)$.

Step2: Calculate area of pre - image

The pre - image $\triangle XYZ$ has vertices $X(3,18)$, $Y(18,18)$, $Z(18,9)$. The base of the triangle (distance between $X$ and $Y$) $b = 18 - 3=15$, and the height (distance from $Z$ to the line $XY$) $h=18 - 9 = 9$. Using the area formula for a triangle $A=\frac{1}{2}bh$, we get $A_{pre - image}=\frac{1}{2}\times15\times9 = 67.5$ square units.

Step3: Calculate area of image

The image $\triangle X'Y'Z'$ has vertices $X'(1,6)$, $Y'(6,6)$, $Z'(6,3)$. The base (distance between $X'$ and $Y'$) $b'=6 - 1 = 5$, and the height (distance from $Z'$ to the line $X'Y'$) $h'=6 - 3=3$. Using the area formula for a triangle $A=\frac{1}{2}bh$, we get $A_{image}=\frac{1}{2}\times5\times3 = 7.5$ square units.

Step4: Find relationship between areas

The ratio of the area of the image to the area of the pre - image is $\frac{A_{image}}{A_{pre - image}}=\frac{7.5}{67.5}=\frac{1}{9}$. In general, if the scale factor of dilation is $k$, the ratio of the area of the image to the area of the pre - image is $k^{2}$. Here $k=\frac{1}{3}$ and $k^{2}=\frac{1}{9}$.

Step5: Calculate perimeter of image

If the perimeter of the pre - image is $P_{pre - image}=41.49$ units and the scale factor of dilation is $k=\frac{1}{3}$, the perimeter of the image $P_{image}$ is related to the perimeter of the pre - image by the formula $P_{image}=kP_{pre - image}$. So $P_{image}=\frac{1}{3}\times41.49 = 13.83$ units.

Answer:

a. Coordinates of $X'=(1,6)$, $Y'=(6,6)$, $Z'=(6,3)$
b. $X'(1,6)$, $Y'(6,6)$, $Z'(6,3)$
c. Area of pre - image: $67.5$ square units, Area of image: $7.5$ square units
d. The area of the image is $\frac{1}{9}$ of the area of the pre - image.
e. Perimeter of the image is $13.83$ units.