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# Check my Proofs

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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.

(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!

Feb 10, 2018

#1
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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   =   k2(xy)   =   k2( Area of Rectangle 1 )

Feb 10, 2018
#2
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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   !!!!

Feb 10, 2018
edited by CPhill  Feb 10, 2018
#3
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"Which would make Rectangle 2 k^2 times the area of Rectangle 1"

Does this mean...

it would make Rectangle   2k2   times the area of Rectangle 1

or

it would make  Rectangle 2   k2 times the area of Rectangle 1

?

That is the question!

Feb 10, 2018