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 original rectangle has sides measuring x inches and y inches, so the area of the original rectangle is A = x * y.
To dilate the rectangle using a scale factor of 1/4, each dimension is multiplied by 1/4. Therefore, the sides of the dilated rectangle will measure (1/4)x inches and (1/4)y inches.
The area of the dilated rectangle is A' = (1/4)x * (1/4)y = (1/16)xy.
To find 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.
(A / A') = (x * y) / ((1/16)xy)
(A / A') = (16/1)
(A / A') = 16
Therefore, the area of the original rectangle is 16 times greater than the area of the dilated rectangle.
In summary:
- The area of the original rectangle is A = x * y.
- The area of the dilated rectangle is A' = (1/16)xy.
- The area of the original rectangle is 16 times greater than the area of the dilated rectangle.