Bunuel wrote:

In the figure, OR and PR are radii of circles. The length of OP is 4. If OR = 2, what is PR ? PR is tangent to the circle with center O.

(A) 2

(B) 5/2

(C) 3

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

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

Attachment:

2017-10-27_1118.png

Angle R is a right angle. PR is tangent to the circle with center O. A tangent to a circle is always perpendicular to the radius of the circle. PR is perpendicular to OR. Hence R is a right angle.

We can infer: this is a 30-60-90 right triangle with side lengths in ratio

\(x: x\sqrt{3}: 2x\)

Rule: If one leg and a hypotenuse of a right triangle are in the ratio of a 30-60-90 triangle, it is a 30-60-90 triangle.*

OR = 2: corresponds with x

OP = 4: corresponds with 2x

PR therefore corresponds with \(x\sqrt{3}\), which = \(2\sqrt{3}\)

Answer D

**

By the Pythagorean theorem, this rule could not be otherwise, so if you recognize the relationship, you do not have to do this math:

Let OR = a

Let PR = b

Let OP = c

\(a^2 + b^2 = c^2\)

\(2^2 + b^2 = 4^2\)

\(b^2 = 16 - 4\)

\(b^2 = 12 = 2\sqrt{3} =

PR\)
_________________

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