Tài liệu Master the Gre 2010 - Part 31 doc

18.

The figure above shows a square that is tangent to one circle at four

points, and inscribed in another. If the diameter of the large circle is 10,

what is the diameter of the smaller circle?

(A) 5=3

2

(B) 5

(C) 2p

(D) 5=2

(E) 7.5

The correct answer is (D). The square’s diagonal is equal in length to the large

circle’s diameter, which is 10. This diagonal is the hypotenuse of a triangle whose

legs are two sides of the square. The triangle is right isosceles, with sides in the

ratio 1:1:=2. The length of each side of the square 5 10

=2

, or 5=2. This length

is also the diameter of the small circle.

Tangents and Inscribed Circles

A circle is tangent to a line (or line segment) if they intersect at one and only one point

(called the point of tangency). Here’s the key rule to remember about tangents: A line

that is tangent to a circle is always perpendicular to the line passing through the

circle’s center at the point of tangency.

The next figure shows a circle with center O inscribed in a square. Point P is one of

four points of tangency. By definition, OP ⊥ AB.

Chapter 11: Math Review: Geometry 283

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