Nested Circles: MGMAT Geometry Question Bank

[hkparikh09]

If P, Q and R are the centers of circles P, Q, and R and the points P, Q, R and T all lie on the same line, what portion of circle P is shaded?

I can't get the image provided in the question to post on this blog. The key to the question was to assume certain lengths for the line segments. How can we assume? There are an infinite number of possibilities?

Do we go off the picture saying, "it looks like line QT is twice the size of line RT"?

Here is the MGMAT explanation:
Let us say that line segment RT has a length of 1. RT is the radius of circle R, so circle R has a radius of 1.

Line segment QT is the diameter of circle R, so it has a length of 2 (twice the radius of circle R). Segment QT also happens to be the radius of circle Q, which therefore has a radius of 2.

Line segment PT, being the diameter of circle Q, has a length of 4. Segment PT also happens to be the radius of circle P, which therefore has a radius of 4.

The question is asking us what fraction of circle P is shaded. The answer will be

(shaded area) ÷ (area of circle P)

The area of circle P is π(4)2, which equals 16π. The shaded area is just the area of circle Q (i.e. π(2)2, which equals 4π) minus the area of circle R (i.e. π(1)2, which equals π). Therefore, the answer to our question is



(4π - π)/16π = 3/16

[RonPurewal]

If you can't embed the image, please post it on an image-hosting site such as postimage.org, and then post a link here.

Thanks.

[hkparikh09]

[RonPurewal]

What are the answer choices?

If the answer choices are fractions, then you can pick any lengths that work.
Remember"”the correct answer is the correct answer. Even if there are a million zillion possibilities, anything that can eliminate four of the five choices is enough.
(This is also the whole principle behind "VIC" / "smart numbers" / "plugging in your own numbers".)

In fact, if the answer choices are five constant fractions, that's even better"”you're absolutely guaranteed that you'll only have to plug in one set of numbers.
(If there are variables in the choices, then it's possible that, by coincidence, more than one choice could survive the first round of plugging.)