DeeDee draws a rectangle with sides measuring x inches and y inches. She dilates the rectangle using a scale factor of 1/4
. What is the area of the original rectangle in terms of x and y? What is the area of the dilated rectangle? How many times greater is the area of the original rectangle compared to the area of the dilated rectangle? Explain your answers and show all your work.
The area of a rectangle is found by multiplying the length by the width.
Therefore, the area of the original rectangle is xy.
To dilate the rectangle using a scale factor of 1/4, each side of the rectangle is multiplied by 1/4. This means that the new sides of the dilated rectangle are (1/4)x and (1/4)y.
So, the area of the dilated rectangle is (1/4)x * (1/4)y = (1/16)xy.
To determine how many times greater the area of the original rectangle is compared to the area of the dilated rectangle, we divide the area of the original rectangle by the area of the dilated rectangle:
Area of original rectangle / Area of dilated rectangle = (xy) / ((1/16)xy)
Dividing by a fraction is the same as multiplying by its reciprocal, so this is equivalent to:
(xy) * (16/xy) = 16
Therefore, the area of the original rectangle is 16 times greater than the area of the dilated rectangle. This makes sense, as dilating a shape by a scale factor less than 1 will always result in a smaller shape, whereas a scale factor greater than 1 will result in a larger shape.