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In the following figure, two circles with center A and B touch a larger circle with center O internally. The ratio of the radii of circle A to circle B is 7:9.

Quantity A: OA

Quantity B: OB

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from information given.

enter image description here

Since the radius of circle B is bigger than the radius of circle A, I concluded that the center point of B is closer to point O which means that OA is greater. However this practice test I'm taking says that the relationship cannot be determined. This practice test has already led me astray a couple times with bad answers, is this another instance of that or why can't we determine which line segment is longer?

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    $\begingroup$ I think it is another bad answer... $\endgroup$ – MattG88 Dec 18 '16 at 23:56
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    $\begingroup$ Your answer is correct, unless there is some silly wordplay on "touch a larger circle ... internally". What bothers me about that is "larger" since in normal speak, if a circle is tangent to another one "internally", then that other one would implicitly have to be larger - so I don't see why that needed to be restated. $\endgroup$ – dxiv Dec 19 '16 at 2:31
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$$\frac{R_A}{R_B}=\frac{7}{9} \Rightarrow R_B>R_A$$

If the circle with center $A$ is internally tangent to the circle with center in $O$ then we now that both centers and the tangent point lies on the same line and then we would have:

$$OA=R_O-R_A$$

If the circle with center $A$ is tangent to the circle with center in $O$ we would have:

$$OB=R_O-R_B$$

And once $R_B>R_A$ we would have $OA>OB$.

The only thing I can see is: "touch internally" doesn't mean "tangent internally", that's why we can't construct the above relations.

However, if they mean "touch internally = tangent internally" then $OA>OB$ and they made some mistake in the statement.

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