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 is k2 times the area of Rectangle 1.
​

Answer 1

(a) Rectangle 1 and rectangle 2 are similar

(b) The perimeter of Rectangle 2 is k times the perimeter of Rectangle 1

(c) The area of Rectangle 2 is k² times the area of Rectangle 1

Explanation:

Answer 2

(a) Rectangle 1 and rectangle 2 are similar

(b) The perimeter of Rectangle 2 is k times the perimeter of Rectangle 1

(c) The area of Rectangle 2 is k² times the area of Rectangle 1

Explanation:

* Lets talk about the similarity

- Two rectangles are similar if there is a constant ratio between

their corresponding sides

- Rectangle 1 has dimensions x and y

- Rectangle 2 has dimensions kx and ky

- The ratio between their dimensions is:

kx/x = k and ky/y = k, so there is a constant ratio K between their

corresponding dimensions

(a) Rectangle 1 and rectangle 2 are similar

- The perimeter of any rectangle is 2(the sum of its two dimensions)

∵ Rectangle 1 has dimensions x and y

∴ Its perimeter = 2(x + y) = 2x + 2y ⇒ (1)

∵ Rectangle 2 has dimensions kx and ky

∴ Its perimeter = 2(kx + ky) = 2kx + 2ky

- By taking k as a common factor

∴ Its perimeter = k(2x + 2y) ⇒ (2)

- From (1) and (2)

∵ The perimeter of rectangle 1 = (2x + 2y)

∵ The perimeter of rectangle 2 = k(2x + 2y)

∴ The perimeter of rectangle 2 is k times the perimeter of rectangle 1

(b) The perimeter of Rectangle 2 is k times the perimeter of Rectangle 1

- The area of any rectangle is the product of its two dimensions

∵ Rectangle 1 has dimensions x and y

∴ Its area = x × y = xy ⇒ (1)

∵ Rectangle 2 has dimensions kx and ky

∴ Its area = kx × ky = k²xy ⇒ (2)

- From (1) and (2)

∵ The area of rectangle 1 = xy

∵ The area of rectangle 2 = k²xy

∴ The area of rectangle 2 is k² times the area of rectangle 1

(c) The area of Rectangle 2 is k² times the area of Rectangle 1