Question

Consider the dilation of square KLMN by a scale factor of 1/4, with a center of dilation at the origin. Assume the units are centimeters.
1. What is the perimeter of the square after the dilation? Explain the change in the perimeter.

2. What is the area of the square after the dilation? Explain the change in the area.

Answer 1

1. The given squares perimeter after dilation is 16 cm. Perimeter is reduced by 48 cm due to dilation.

2. The squares area is 16 square centimeters after dilation. The area reduces by 240 square cm due to dilation

Step-by-step explanation:

Step 1; First, we determine the coordinates of the points given in the graph. By plotting we determine that the points are; K (-8, -8), L(8,-8), M(8, 8) and N(-8, 8).

Step 2; We calculate the new coordinates of the square KLMN by multiplying the scale factor with each of the coordinate's x and y values.

8 × $\frac{1}{4}$ = 2, -8  $\frac{1}{4}$ = -2 are the different values in the four coordinates.

So by multiplying the scale factor we get the dilated square's points as

K (-2, -2), L(2,-2), M(2, 2) and N(-2, 2).

Step 3; The perimeter of a square is given by 4 times its side length. As the units are in centimeters, we determine the side length is 4 cm whereas it is 16 cm for the undilated square.

Perimeter of original square = 4 × 16 cm = 64 cm

Perimeter of the dilated square = 4 × 4 cm = 16 cm.

Change in perimeter = Perimeter of original square - Perimeter of the dilated square = 64 cm - 16 cm = 48 cm.

Step 4; The area of any given square is the side length of the square squared. The side length of the original square is 16 cm whereas the side length of the dilated square is 4 cm.

Area of original square = 16 × 16 cm = 256 square cm

Area of the dilated square = 4 × 4 cm = 16 square cm.

Change in area = area of original square - area of the dilated square = 256 cm - 16 cm = 240 square cm.