Suppose that a triangle with sides
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$\,a\,$,
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$\,b\,$, and
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$\,c\,$ has
been scaled by a positive number
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$\,s\,$
to get a similar triangle
with corresponding sides
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$\,A\,$,
$\,B\,$, and
$\,C\,$.
Thus,
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$\,A = sa\,$ and
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$\,B = sb\,$ and
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$\,C = sc\,$.
Let's investigate the relationship between the perimeters of these two triangles:
[beautiful math coming... please be patient] $\text{perimeter of original triangle} = a + b + c\,$
Thus, the perimeter ends up being scaled by the same factor that scales the sides!
This can be rephrased in terms of ratios.
Notice that:
$$
\frac{\text{scaled perimeter}}{\text{original perimeter}}
= \frac{A+B+C}{a+b+c}
= \frac{sa+sb+sc}{a+b+c}
= \frac{s(a+b+c)}{a+b+c}
= s = \text{scaling factor}
$$
and
$$\frac{\text{scaled side}}{\text{original side}} = \frac{A}{a} = \frac{sa}{a} = s$$
$$\frac{\text{scaled side}}{\text{original side}} = \frac{B}{b} = \frac{sb}{b} = s$$
$$\frac{\text{scaled side}}{\text{original side}} = \frac{C}{c} = \frac{sc}{c} = s$$
So, the ratio of the perimeters is the same as the ratio of corresponding sides.
A similar calculation shows that this result is indeed true for polygons in general:
Area, on the other hand, behaves quite differently.
Suppose you have a square of side
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$\,a\,$
that has been scaled by
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$\,s\,$
to get a square of side
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$\,A\,$. Let's investigate the relationship between the areas of these two squares: [beautiful math coming... please be patient] $\text{area of original square} = a\cdot a = a^2$
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$\text{area of scaled square}$
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$= A\cdot A$
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$= (sa)\cdot(sa)$
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$= s^2\cdot a^2$
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$= s^2(\text{area of original square})$
Thus, the area ends up being scaled by the square of the factor that scales the sides! 
Again, this can be rephrased in terms of ratios.
Notice that:
$$
\frac{\text{scaled square area}}{\text{original square area}}
= \frac{A\cdot A}{a \cdot a}
= \frac{(sa)\cdot (sa)}{a\cdot a}
= \frac{s^2a^2}{a^2}
= s^2 = \text{the square of the ratio of corresponding sides}
$$
This idea leads to the following theorem:
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
