b Graph the rectangle after dilation in a coordinate system.
C
c Graph the rectangle after dilation in a coordinate system.
D
d Compare the scale factors with the perimeters and areas of the dilated figures.
A
a P=24 units, A=32 square units
B
b P=72 units, A=288 square units. The perimeter is 3 times the original and the area is 9 times.
C
c P=6 units, A=2 square units. The perimeter is 14 times the original and the area is 116 times.
D
d The perimeter is scaled by a factor k and the area of a factor k^2.
Practice makes perfect
a To find the perimeter and the area we will graph the rectangle in a coordinate system.
Now we need to find the base and the height of the rectangle to calculate the perimeter and the area. We can measure them in the coordinate system.
The base of the rectangle is 8 units and the height is 4 units. The perimeter is the sum of all the sides in the rectangle. Therefore, we should add two times the base and two times the height to get the perimeter, P.
P=8+8+4+4=24
The perimeter is 24 units and we will now calculate the area. We will multiply the base of the rectangle with the height to get the area, A.
A=8*4=32
The area of the rectangle is therefore 32 and the unit is square units since it's an area.
b We will multiply the coordinates of all vertices with the scale factor k=3 to get the points after dilation.
(x,y) &→ (3x,3y)
W(-3,-1) &→ W'(-9,-3)
X(-3,3) &→ X'(-9,9)
Y(5,3) &→ Y'(15,9)
Z(5,-1) &→ Z'(15,-3)
We can now draw the image in the same coordinate system as the original rectangle.
The base and the height of the dilated rectangle can now be found by calculating the units between the points in the picture or by using the Distance Formula. Let's use the Distance Formula and start with the base, the distance between W' and Z'. We will use the coordinates for W'(-9,-3) and Z'(15,-3) in the formula.
The base is therefore 24 units and we will now calculate the height, the distance between W' and X'. Insert the coordinates for W'(-9,-3) and X'(-9,9) in the Distance Formula.
The height is 12 units and together with the base we can calculate the perimeter and the area. The perimeter of a rectangle is the sum of all sides, two times the base and two times the height. The perimeter P will therefore be:
P=24+24+12+12=72.
The perimeter is 72 units and we will now calculate the area, A, by multiplying the base with the height.
A=24*12=288
We should now compare the perimeter and area with the original rectangle. We will do that by calculating the ratio between them, dividing the dilated perimeter with the original perimeter and the same for the area.
Dilated rectangle
Original rectangle
Ratio
Perimeter
72
24
3
Area
288
32
9
The dilated perimeter is 3 times the original one and the dilated area is 9 times the original area. Since each length is scaled with 3 the difference when multiply them becomes 3*3=9.
c We should now multiply the vertices of the rectangle WXYZ by the scale factor k= 14. Multiplying by 14 is the same thing as dividing by 4.
(x,y) &→ ( x4, y4)
W(-3,-1) &→ W' (- 34,- 14)
X(-3,3) &→ X' (- 34, 34)
Y(5,3) &→ Y' ( 54, 34)
Z(5,-1) &→ Z' ( 54,- 14)
With the new vertices we can graph the image of the dilation in the same coordinate system as the original rectangle. It's a bit tricky since the coordinates are fractions and not integers. For example 34 should be placed between 0 and 1 but closer to 1.
Since it's hard to see the length of the base and height from the graph we will use the Distance Formula to calculate them. We will start with the base and use the coordinates W' (- 34,- 14) and Z' ( 54,- 14).
The base of the dilated rectangle is therefore 2 units. Now we will do the same thing for the height with the coordinates W' (- 34,- 14) and X' (- 34, 34).
The height of the dilated rectangle is therefore 1 unit. To calculate the perimeter P we will add two times the base and two times the height.
P=2+2+1+1=6.
The perimeter is 6 units and to calculate the area A we will multiply the base with the height:
A=2*1=2.
Finally, we should should compare the perimeter and the area of the dilated rectangle with the original one. We will do that by dividing them with each other.
Dilated rectangle
Original rectangle
Ratio
Perimeter
6
24
1/4
Area
2
32
1/16
The dilated perimeter is 14 the size of the original and the dilated area is 116 the size of the original area.
d To make a conjecture we will compare the scale factors with the perimeters and areas of the dilated figures.
Scale factor
Ratio perimeter
Ratio area
k=3
3
9
k= 14
k= 14
k= 116
We can see that the perimeter is scaled with the same factor as the scale factor k. For the area it's not that clear. When the scale factor is 3, the area ratio is greater and when the scale factor is 14, the area ratio is less than the scale factor. We will search for a pattern and can see that the area ratio is the square of the scale factor.
3*3=9, 14* 14= 116
Therefore, the area is scaled by the square of the scale factor, k^2.
