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

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Math Expert
Joined: 02 Sep 2009
Posts: 42305

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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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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) $$3 \sqrt{2}$$

[Reveal] Spoiler:
Attachment:

2017-10-27_1118.png [ 9.82 KiB | Viewed 513 times ]
[Reveal] Spoiler: OA

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Kudos [?]: 133072 [0], given: 12403

Intern
Joined: 08 Apr 2017
Posts: 29

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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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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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Director
Joined: 22 May 2016
Posts: 998

Kudos [?]: 347 [0], given: 594

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

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27 Oct 2017, 08:47
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) $$3 \sqrt{2}$$

[Reveal] 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 = $$2\sqrt{3}$$

**
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

$$a^2 + b^2 = c^2$$
$$2^2 + b^2 = 4^2$$
$$b^2 = 16 - 4$$
$$b^2 = 12 = 2\sqrt{3} = PR$$

Kudos [?]: 347 [0], given: 594

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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