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A square has a perimeter of 28 cm.  All dimensions are increased by a scale factor of 3 to create a similar figure.  What is the area of the original figure? Explain how you would do it.

Guest Feb 27, 2018
 #1
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The original figure has a perimeter of 28 centimeters. In this particular situation, the information regarding the scale factor is extraneous information; simply disregard it; the question asks for the area of the original figure--not the new figure.

 

A square, by definition, is an equilateral quadrilateral, so every side of this four-sided polygon is of equal length. Let's set one of the side lengths of the square as "s."

 

Since the polygon has four sides of equal length, every side has a length of "s." "4s" represents the total area of a generic square. We know the total perimeter of this current square: 28 centimeters. We can use this information to figure out the length of one side.

 

\(4s=28\\ s=7\text{cm}\)

 

The area of a square is the length of its base multiplied by the length of the height. 

 

\(A_{\fbox{}}=7\text{cm}*7\text{cm}=49\text{cm}^2\)

 

Of course, the area is a square unit, so the answer should be in that format. 

TheXSquaredFactor  Feb 27, 2018
 #2
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So the answer would be 49cm^2? So I would I be right if I said "28  divided by 4 is 7, multply that by 7 gives me 49m^2?"

Guest Feb 27, 2018
 #3
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That is the simplified version of my explanation above. It may be a typo on your end, but just be sure that 49m^2 is 49cm^2 

TheXSquaredFactor  Feb 27, 2018
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Yes I meant cm^2. I have one more question for you. Same perimeter of 28cm and scale factor of 3 but now its the new area of the new figure.

Guest Feb 27, 2018
 #5
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This explains the relevance of the scale factor! Let's think about how a scale factor affects the area of a figure before trying to solve. 

 

Since the original square has dimensions 7cm by 7cm, the scale factor would have to affect both the length and the height. Therefore, the length and the height would be tripled; thus, the area would be affected by the square of scale factor since the scale factor is being applied to both dimensions. 

 

We already know the area of the original figure; it is \(49\text{cm}^2\). If the scale factor affects the area by its square, then the area will be enlarged by \(3^2\) or 9.

 

\(49\text{cm}^2*9=441\text{cm}^2\)

 

This is the area of the new square.

TheXSquaredFactor  Feb 27, 2018

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