Question

A rectangle has a base of 24 cm and a height of 33 cm. If it is scaled by 1/3, find the ratio of the perimeters and areas of the smaller rectangle to that of the larger one.

Answer 1

The ratio of the perimeters of the smaller and larger rectangles is 1:3, while the ratio of the areas is 1:9.

Step-by-step explanation:

To find the ratio of the perimeters of the smaller and larger rectangles, we need to calculate the perimeters of both rectangles. The perimeter of a rectangle is given by the formula P = 2(l + w), where l is the length and w is the width.

For the larger rectangle, the length is 33 cm and the width is 24 cm, so the perimeter is P = 2(33 + 24) = 2(57) = 114 cm.

For the smaller rectangle, we scale both the length and width by a factor of 1/3. So the length becomes 1/3 * 33 = 11 cm and the width becomes 1/3 * 24 = 8 cm. Therefore, the perimeter of the smaller rectangle is P = 2(11 + 8) = 2(19) = 38 cm.

The ratio of the perimeters of the smaller and larger rectangles is 38 cm : 114 cm, which can be simplified to 1 : 3.

To find the ratio of the areas of the smaller and larger rectangles, we need to calculate the areas of both rectangles. The area of a rectangle is given by the formula A = l * w.

For the larger rectangle, the length is 33 cm and the width is 24 cm, so the area is A = 33 * 24 = 792 cm^2.

For the smaller rectangle, we scale both the length and width by a factor of 1/3. So the length becomes 1/3 * 33 = 11 cm and the width becomes 1/3 * 24 = 8 cm. Therefore, the area of the smaller rectangle is A = 11 * 8 = 88 cm^2.

The ratio of the areas of the smaller and larger rectangles is 88 cm^2 : 792 cm^2, which can be simplified to 1 : 9.