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In the figure, OR and PR are radii of circles. The length of OP is 4.

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In the figure, OR and PR are radii of circles. The length of OP is 4. [#permalink]

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New post 27 Oct 2017, 00:20
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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) \(3 \sqrt{2}\)


Spoiler: ::
Attachment:
2017-10-27_1118.png
2017-10-27_1118.png [ 9.82 KiB | Viewed 681 times ]
Spoiler: OA

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Re: In the figure, OR and PR are radii of circles. The length of OP is 4. [#permalink]

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New post 27 Oct 2017, 02:33
Tangent to the circle creates a right angle to the radius of the circle.

Hence angle ORP is 90.

By using Pythagoras theorem,
We have OR^2 +PR^2 = OP^2

This gives option D


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In the figure, OR and PR are radii of circles. The length of OP is 4. [#permalink]

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New post 27 Oct 2017, 08:47
Bunuel wrote:
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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) \(3 \sqrt{2}\)

Spoiler: ::
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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In the figure, OR and PR are radii of circles. The length of OP is 4.   [#permalink] 27 Oct 2017, 08:47
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In the figure, OR and PR are radii of circles. The length of OP is 4.

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