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.

(a) Are Rectangle 1 and Rectangle 2 similar? Why or why not?
(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 ismtimes the area of Rectangle 1.

Answer 1

(a) Rectangle 1 and Rectangle 2 are similar rectangles because the lengths of their respective sides are proportional, and their four angles are equal.

(b) The perimeter of a rectangle is the sum of twice its base plus twice its height. For rectangle 1, the perimeter is
`P = 2x + 2y`. For rectangle 2, the perimeter is
`2kx + 2ky = k (2x + 2y) = kP`.

(c) The area of a rectangle is the product between the value of its base and its height. For rectangle 1, the area is
`A = xy`. For rectangle 2, the area is
`(kx) (ky) = k ^ 2 (xy) = k ^ 2A`.

Explanation:

Answer 2

(a) The Rectangles are similar because all angles of all rectangles are equal to 90°. Then, the corresponding sides have the equivalent ratio equal to k.
(b) The perimeter of the rectangle is the sum of the measurements of all sides. Such that for Rectangle 1, it should be.
Perimeter (Rectangle 1) = 2x + 2y
Then for rectangle 2,
Perimeter (Rectangle 2) = 2kx + 2ky = k (2x + 2y)
= k(Perimeter of rectangle 1)

c. Area of rectangle is the product of the lengths of two sides, (x)(y). For Rectangle 2, that would be (kx)(ky) = k²xy