ShmoopTube

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Here’s your shmoop du jour, brought to you by a midpoint.

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Apparently, it has some kind of restraining order against the circumference.

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F is the midpoint of DG and E is the midpoint of FD.

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What is the ratio of the area of the circle with center E to the area of the circle with center G?

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Here are the potential answers...

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Okay, so we’ve got three circles here…and we want to know how many of the itty-bitty

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circles we could fit inside the big circle.

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Well, all right…to be more technically accurate, we want a ratio of the area.

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And all we can use to get there is a couple of midpoints.

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These guys had better be good. Okay… here’s another problem where we’re

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going to have to go ahead and just call something “x.”

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In this case, we know that ED is a radius of the smallest circle… so we’ll call

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that guy “x.”

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That would make FD, the smallest circle's diameter, 2x.

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And, because FD is also a radius of the medium circle…GD is going to be 4x.

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Now that we have our radii, it’s time to whip out the formula for the area of a circle,

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and go to work… For our smallest circle, we’ll take the formula

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Area equals pi times r squared and plug in our… r…

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…to get pi x squared.

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For the big circle, we get pi times the quantity 4x… squared… or pi sixteen x squared.

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We’re looking for the ratio, which is always the same as a fraction…

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…and in this case is pi times x squared over pi times sixteen x squared.

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The pi and the x squared cancel out…leaving us with just 1 over 16…

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…which is equivalent to option A.

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