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Re: In the figure above, the circles are centered at O(0, 0) and
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09 Apr 2022, 04:57
OE:
Before dealing with either statement, note that the line containing A and B is perpendicular to the radii OA and PB of the two circles, creating triangles OAX and PBX. Each triangle contains a right angle, and the angles at point X are also identical (since they are vertical angles). Therefore, triangles OAX and PBX are similar.
(1) SUFFICIENT: Let R stand for the radius of the larger (right-hand) circle, and r the radius of the smaller (left-hand) circle. Then, according to this statement, πR2 = 4πr2. Therefore, R2 = 4r2, and so R = 2r. In other words, PB = 2(OA), or, equivalently, PB/OA = 2.
Since the triangles are similar, the ratio of similar sides is equal: PB/OA = PX/OX. Therefore PX/OX also equals 2. The problem indicates that the length of OP is 10, so this is enough information to determine the lengths of PX and OX, which will allow you to determine the x-coordinate of X.
Don’t actually do this math, but here’s how: OP is 10 and the portion PX is twice as long as the portion OX. The length of PX, then, must be (2/3)(10) while the length of OX must be (1/3)(10). The x-coordinate of X, then, equals 10/3.
(2) SUFFICIENT: Since the triangles are similar, the ratio of similar sides is equal: BX/AX = PX/OX. According to this statement, BX/AX is 2, so PX/OX also equals 2. The problem indicates that the length of OP is 10, so this is enough information to determine the lengths of PX and OX, which will allow you to determine the x-coordinate of X.
Don’t actually do this math, but here’s how: OP is 10 and the portion PX is twice as long as the portion OX. The length of PX, then, must be (2/3)(10) while the length of OX must be (1/3)(10). The x-coordinate of X, then, equals 10/3.
The correct answer is (D).