Question

2. 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.

(a) Are Rectangle 1 and Rectangle 2 similar? Why or why not?
Yes, they are similar because rectangle 2 is a larger version of rectangle 1.
(b) Write a paragraph proof to show that the perimeter of Rectangle 2 is k times the perimeter of Rectangle 1.

(c) Write a paragraph proof to show that the area of Rectangle 2 is times the area of Rectangle 1.

Answer 1

We can say that the polygons are similar if and only if their corresponding sides are equal and that their corresponding angles are congruent.

(a) All rectangles have angles equal to 90° and there exists a factor for the corresponding sides. Therefore, the rectangles are similar.

(b) The perimeter of the rectangle is the sum of all the sides. If we let x and y be the sides then,
P = 2x + 2y
for the second rectangle,
P = 2kx + 2ky = 2k(x + y) = k times perimeter 1

(c) The area of the rectangle is the product of the two sides. For the first rectangle,
A = xy
for the second,
A = (kx)(ky) = k²xy = k² times the area of 1