DeeDee draws a rectangle with sides measuring x inches and y inches. She dilates the rectangle using a scale factor of 14
. 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 original rectangle? Explain your answers and show all your work.
The original rectangle has sides measuring x inches and y inches, so its area is given by A = x * y.
After dilating the rectangle with a scale factor of 14, the new dimensions of the rectangle will be 14x and 14y. Therefore, the area of the dilated rectangle is A' = (14x) * (14y) = 196xy.
To find how many times greater the area of the original rectangle is compared to the area of the dilated rectangle, we can divide the area of the original rectangle by the area of the dilated rectangle: A / A' = (x * y) / (196xy) = 1/196.
Therefore, the area of the original rectangle is 1/196 times the area of the dilated rectangle.
Explanation: Dilating a shape with a scale factor multiplies its dimensions by that scale factor. Since area is the product of length and width, the area will be multiplied by the square of the scale factor. In this case, the scale factor is 14, so the area of the dilated rectangle will be 14^2 = 196 times greater than the area of the original rectangle.