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# Similar shapes

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May 16, 2018

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In order for a pair of polygons to be considered similar, the polygon must meet two conditions:

1) All corresponding angles are congruent

2) The ratio of corresponding sides of both polygons is equal (also known as constant of proportionality)

Since the ratio of the side lengths is equal, we can set up a proportion to represent the equal ratios.

 $$\frac{x}{x+2}=\frac{4x}{48}$$ The right-hand fraction can be simplified, which should make the further calculations easier to do. $$\frac{x}{x+2}=\frac{x}{12}$$ When dealing with proportions, cross-multiplying is generally considered most efficient because you are eliminating all fractions. $$12x=x(x+2)$$ Let's distribute on the right hand side. $$12x=x^2+2x$$ Let's subtract 12x from both sides. $$x^2-10x=0$$ This quadratic does not have a "c" term, so it is easiest to factor out an "x." $$x(x-10)=0$$ Set both factors equal to zero and solve. $$x_1=0\text{cm}\\ x_2=10\text{cm}$$ As geometrists, we must reject x1 as this would result in a side length with length 0, which does not make any sense. We are not done yet, though! We must solve for the width. $$w=4x=4*10=40\text{cm}$$
May 16, 2018