Question

Rectangle 1 has length x and width y. Rectangle 2 is made by multiplying each dimension of Rectangle 1 by a factor of k, where k > 0. Are Rectangle 1 and Rectangle 2 similar? Why or why not? Write a paragraph proof to show that the perimeter of Rectangle 2 is k times the perimeter of Rectangle 1. Write a paragraph proof to show that the area of Rectangle 2 is times the area of Rectangle 1. Answer:

Answer 1

Yes, the two rectangles are similar, because rectangle 2 is a dilation of rectangle 1.

Are the two rectangles similar?

We know that rectangle 1 has dimensions L and W.

And rectangle 2 is made by multiplying the dimensions of rectangle 1 by a factor k > 0.

Then, rectangle 2 is just a dilation of rectangle 1, this means that in fact, the two rectangles are similar by definition.

Then:

Dimensions of rectangle 1:

• Length = L
• Width = W.
• Perimeter = 2*(W + L)
• Area = W*L

For rectangle 2:

• Length = k*L
• Width = k*W
• Perimeter = 2*(k*L + k*W) = k*(2*(L + W))
• Area = (k*L)*(k*W) = k²(L*W)

Above we can see that the perimeter of rectangle 2 is k times the perimeter of rectangle 1, and the area of rectangle 2 is k squared times the area of the rectangle 1.

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