Bunuel wrote:

The circle shown is tangent to both the x and y-axes. If the length of the segment from the circle’s center C to the origin, O, is 6, what is the circle’s radius?

(A) 6

(B) \(3 \sqrt{2}\)

(C) \(2 \sqrt{3}\)

(D) 3

(E) 2

Attachment:

2018-04-01_2137.png

We can drop a perpendicular from point C to the x-axis, and call the point of intersection D so that triangle OCD is a 45-45-90 triangle (note: CD is a radius of the circle that is perpendicular to OD which is lying on the x-axis). We will use the fact that the ratio of a side to the hypotenuse of a 45-45-90 triangle is x : x√2.

If we let CD = OD = x = the length of (either) side of the 45-45-90 triangle, then the length of the hypotenuse of the 45-45-90 triangle is x√2. From the diagram, we know that the length of the hypotenuse is 6. Thus, we can create the equation:

x√2 = 6

x = 6/√2

We need to rationalize the denominator. Multiplying by √2/√2, we have:

x = 6√2/2 = 3√2

Answer: B

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Jeffery Miller

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