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A rectangle has an area of x square units. The rectangle is dilated to create a new rectangle with an area of 9x square units. Describe the change in the dimensions of the rectangle.

-What is the scale factor used in the dilation?

-If the demsions of the original rectangle are 5 inches by 8 inches, what will be the dimensions of the new rectangle? Show your work.

Sagot :

TSOidn
The reactangle has an area of x.

So that means two numbers multiplied to each other equal to x. We can just think of itwo numbers that when multiplied equal to x as a and b. 

So that means the side lengths are a and b. And ab = x (will be using this right next).

And it was multiplied by some scale factor that made the new area 9x. So if the scale factor is y, then:

[tex](y\times a)\times (y\times b)=9x[/tex]
[tex]y^2ab=9x[/tex]
[tex]y^2x=9x[/tex]
[tex]y=3[/tex]

y can not equal to -3, because the length of a side of any polygon can never be negative.

So the dilation is 3.

And so if the dimension is 5 by 8.

Then the area is [tex]5 \times 8=40[/tex]

And so then [tex]x=40[/tex]

And then after the dilation, it would be [tex]9x=9(40) = 360[/tex]

We could check it another way too. Since we said the scale factor is 3, then:

[tex]( 5 \times 3) \times (8 \times 3) = 15 \times 24 = 360[/tex]

So basically 15 by 24 is the new dimensions :)

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