ShmoopTube

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Here’s your shmoop du jour, brought to you by squares within squares.

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It’s like a spiraling S’more vortex.

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ABCD is a square.

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Points A prime, B prime, C prime, and D prime are constructed by translating the points

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A, B, C and D over two-thirds the length of the side.

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If the perimeter of A prime B prime C prime D prime is

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8 times the square root of 5, what is the perimeter of ABCD?

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Here are the potential answers...

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We’ve got one square – ABCD. Inside that square, we’ve got a slightly smaller one…

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that we’ve created by plunking down four new points along the first square’s edges.

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There are really only two juicy bits of information we’re given.

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First, that the new points, which we call “prime,” are translated from their original

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locations 2/3 of the way along each side.

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And second, that the perimeter of that inner circle is 8 square root of 5.

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All righty…the easiest piece to extrapolate is the perimeter of the inner square.

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We know that square has 4 equal sides…so if the total perimeter is 8 square root of

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5, then the length of each side must be exactly 1/4 of that…or 2 square root of 5.

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The next thing we should notice is that our inner square has magically created some triangles…

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We’re not given any measurements for the outer square, but we can always call one

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side “x”, so… let’s do that.

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Because we’re told where all those “prime” points are, we know that one side of our triangle

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is going to be 1/3x and the other side is going to be 2/3x.

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We now have ways to reference all three sides of one of our triangles.

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So…let’s Pythagorize ‘em…

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1/3x squared plus 2/3x squared equals 2 square root of 5 squared.

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We get 1/9x squared plus 4/9x squared equals 4 times 5.

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Simplifying further, 5/9x squared equals 20.

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We can now multiply both sides by 9 to get 5x squared equals 180…

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…and then divide both sides by 5 to get x squared equals 36.

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Last step – take the square root of both sides, and x equals 6.

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Now…be careful – that’s only the length of one side, and we’re looking for the entire perimeter.

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So…just multiply that 6 by 4 and we get there. The answer is 24 or E. Done.

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