consider pairs of rectangles where the dimensions are doubled (rectangles with all sides X2 and all sides X4, or those with all sides X3 and all sides X6. what happens to the perimeter when you double each of the dimensions of a rectangle?, Then what happens to the area? Did perimeter and area of rectangles change in the same way it did for squares?

If x and y are the two sides of any rectangle/square, then the perimeter = 2x + 2y. The area = x * y.

When doubled, the perimeter = 4x + 4y and the area = 2x * 2y.

Use this analogy to answer your questions.

Which of the following rectangles has the same perimeter as the one shown above?

A.
a rectangle with a height of 5 cm and width of 3 cm

B.
a rectangle with a height of 9 cm and width of 2 cm

C.
a rectangle with a height of 7 cm and a width of 3 cm

D.
a rectangle with a height of 8 cm and a width of 5 cm

When you double each of the dimensions of a rectangle, such as doubling the length and width, the perimeter of the rectangle also doubles. This is because the perimeter of a rectangle is the sum of all its sides. So, if each side is multiplied by 2, the total perimeter will also be multiplied by 2.

To visualize this, consider a rectangle with length 'L' and width 'W'. The perimeter (P) of this rectangle is given by the formula: P = 2L + 2W.

When you double each of the dimensions (2L and 2W), the new perimeter becomes:
P' = 2(2L) + 2(2W) = 4L + 4W = 4(L + W).

As you can see, the new perimeter is four times the original perimeter.

Now, let's move on to the area of the rectangle. The area of a rectangle is calculated by multiplying its length and width. So, if each dimension is doubled, the new area becomes four times the original area.

Using the same rectangle with length 'L' and width 'W', the area (A) is given by the formula: A = L * W.

When you double each of the dimensions (2L and 2W), the new area becomes:
A' = (2L) * (2W) = 4(L * W) = 4A.

Therefore, when you double each of the dimensions of a rectangle, the area is multiplied by a factor of 4.

To summarize:
- The perimeter of a rectangle doubles when you double each of its dimensions.
- The area of a rectangle is multiplied by a factor of 4 when you double each of its dimensions.

Regarding your question about squares, the changes in perimeter and area are the same for squares when the dimensions are doubled. The perimeter of a square is given by P = 4S, where S is the length of a side. When you double the side of a square, the new perimeter becomes 4(2S) = 8S, which is twice the original perimeter. Similarly, the area of a square is given by A = S^2, where S is the length of a side. When you double the side of a square, the new area becomes (2S)^2 = 4S^2, which is four times the original area.