a We are asked to draw polygon WXYZ and then find its perimeter. First, let's plot the vertices of the figure on the coordinate plane and then connect them. This will give us the graph of WXYZ.
To calculate the perimeter, we need to find the length of each side of WXYZ. We will use the Distance Formula for this. Let's substitute the coordinates of points W(6,2) and X(3,7) to find the length of WX.
Following similar steps, we can calculate the lengths of the other sides.
Sides
Points
Substitution
Calculation
WX
W (6,2) & X (3,7)
WX = sqrt(( 3- 6)^2+( 7- 2)^2)
WX = sqrt(34) ≈ 5.8
XY
X (3,7) & Y (- 1,4)
XY = sqrt(( - 1- 3)^2+( 4- 7)^2)
XY = sqrt(25) = 5
YZ
Y (- 1,4) & Z (4,- 2)
YZ = sqrt(( 4-( - 1))^2+( - 2- 4)^2)
YZ = sqrt(61) ≈ 7.8
ZW
Z (4,- 2) & W (6,2)
ZW = sqrt(( 6- 4)^2+( 2-( - 2))^2)
ZW = sqrt(20) ≈ 4.5
The sum of these lengths is the perimeter P of polygon WXYZ.
P & = WX + XY + YZ + ZW
& = 5.8+ 5 + 7.8+ 4.5
& = 23.1
The perimeter of WXYZ is 23.1 units.
b This time we want to graph the image of WXYZ after a dilation of 12=0.5 centered at the origin. To find the vertices of the image, we multiply the x- and y-coordinates of each vertex by the scale factor k=0.5.
lcl
(x,y) & → & (0.5x, 0.5y) [0.6em] [-0.6em]
W(6,2) & → & W'(3,1) [0.5em]
X(3,7) & → & X'(1.5,3.5) [0.5em]
Y(- 1,4) & → & Y'(- 0.5,2) [0.5em]
Z(4,- 2) & → & Z'(2,- 1)
Now we will plot the points and connect them. The result is the dilated image.
c Our goal is to find the perimeter of the dilated image from Part B. Same as before, we can use the Distance Formula to find the length of each side of the dilated image W'X'Y'Z'. Let's start by calculating W'X'.
The perimeter P_d of the dilated image is 11.6 units.
P_d & = 2.9+ 2.5 + 3.9+ 2.2
& = 11.5
Now let's calculate the ratio between the perimeter P_d of the dilated image and the perimeter P of WXYZ.
P_d/P &= 11.5/23.1
&≈ 0.5
See that the result is about half the perimeter of the original image. The approximation is a result of rounding the values that were used to calculate the perimeters.
