Original Question:

1. 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 k^2 times the area of Rectangle 1.

Answers for (b) and (c):

(b):

(b) The perimeter of the rectangle is the sum of the measurements of all sides. Rectangle 1 should be, 2(W+L)=2(x+y). Rectangle 2 should be, 2(W+L)=2(2x+2y)=2(2(x+y)=4(x+y)=2[2(x+y)]. Which then simplifies to 2 times the perimeter of Rectangle 1.

(c) The area of a rectangle is the product of the lengths of two sides. To find Rectangle 1’s area you would multiply (x)(y). For Rectangle 2 the area formula would be (kx)(ky)=k^2*xy. Which would make Rectangle 2 k^2 times the area of Rectangle 1.

My other paragraph proof for (b) was...But didn't know if it as correct.:

The perimeter of the rectangle is the sum of the measurements of all sides. Rectangle 1 should be, Perimeter(Recangle 1)=2x+2y. Rectangle 2 should be, Perimeter(Recangle 2)=2kx+2ky. Which then simplifies to 2 times the k(2x+2y)=k(Perimeter of Rectangle 1).

Thank You!

KennedyPape
Feb 10, 2018

#1**+1 **

On your first paragraph proof for (b) , you said, "Rectangle 2 should be, 2(W+L)=2(2x+2y)"

....Where did you get that the width and length of Rectangle 2 are 2x and 2y ?

Rectangle 1 has length x and width y .

Rectangle 2 has a length kx and a width ky .

Perimeter of Rectangle 1 = x + x + y + y

Perimeter of Rectangle 2 = kx + kx + ky + ky = k(x + x + y + y) = k( Perimeter of Rectangle 1 )

Your second paragraph proof for (b) is correct.

And your paragraph proof for (c) is also correct.

Area of Rectangle 1 = xy

Area of Rectangle 2 = (kx)(ky) = k*x*k*y = k*k*x*y = k^{2}(xy) = k^{2}( Area of Rectangle 1 )

hectictar
Feb 10, 2018

#2**+1 **

Thanks, hectictar.....here's my input.....

C looks good except for the last part.....it should be this :

" For Rectangle 2 the area formula would be (kx)(ky)=k^2*xy. Which would make Rectangle k^2 times the area of Rectangle 1. "

[We don't need the "2" ]

For B....consider this

The perimeter of the original rectangle = ( 2x + 2y) . And the sides of the "new rectangle" are kx and ky. So its perimeter is 2 (kx) + 2(ky) = k (2x + 2y) = "k" times the perimeter of the first.

Hope that helps !!!!

CPhill
Feb 10, 2018