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# The circle in the figure above has center O. Which of the following

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Math Expert
Joined: 02 Sep 2009
Posts: 44373
The circle in the figure above has center O. Which of the following [#permalink]

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20 Nov 2017, 23:15
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The circle in the figure above has center O. Which of the following measures for the figure would be sufficient by itself to determine the radius of the circle?

I. The length of arc PQR
II. The perimeter of ∆ OPR
III. The length of the chord PR

(A) None
(B) I only
(C) II only
(D) III only
(E) I, II and III

[Reveal] Spoiler:
Attachment:

2017-11-21_1003_001.png [ 6.69 KiB | Viewed 585 times ]
[Reveal] Spoiler: OA

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The circle in the figure above has center O. Which of the following [#permalink]

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20 Nov 2017, 23:50
Bunuel wrote:

The circle in the figure above has center O. Which of the following measures for the figure would be sufficient by itself to determine the radius of the circle?

I. The length of arc PQR
II. The perimeter of ∆ OPR
III. The length of the chord PR

(A) None
(B) I only
(C) II only
(D) III only
(E) I, II and III

[Reveal] Spoiler:
Attachment:
2017-11-21_1003_001.png

I. Length of arc PQR = (90/360) * 2*pi*r
II. Perimeter of OPR = 2*r + \sqrt{2}*r
III. Lenght of Chord PR = \sqrt{2}*r.. Hence, E.
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Joined: 29 Aug 2016
Posts: 32
Re: The circle in the figure above has center O. Which of the following [#permalink]

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20 Nov 2017, 23:54

1) Length of the arc is known. So we know that the sector here is a whole quadrant. As the length of the arc is related to the angle 'X' subtended by the equation:

Length of arc = (X/360) X Circumference.

Circumference is 2(pi)(r).

2) If we know the perimeter, say P.
It is an isosceles Triangle as it is clear from the right angle and the diagram.
So sides OP and OR are equal, say x units. We can deduce PR from the equation of perimeter. And other equation can be the pythagorean theorem for the Triangle. Hence solved.

3) Length of the cord PR pretty much tells all the lengths involved. Hence the radius can be easily found out using trigonometry.

So all these statements can be used to find the radius.

Hence E.

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Joined: 22 May 2016
Posts: 1427
The circle in the figure above has center O. Which of the following [#permalink]

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21 Nov 2017, 12:29
Bunuel wrote:

The circle in the figure above has center O. Which of the following measures for the figure would be sufficient by itself to determine the radius of the circle?

I. The length of arc PQR
II. The perimeter of ∆ OPR
III. The length of the chord PR

(A) None
(B) I only
(C) II only
(D) III only
(E) I, II and III

[Reveal] Spoiler:
Attachment:
2017-11-21_1003_001.png

Assess options: Enough to determine radius?

My analysis does not involve trig.

There is an isosceles right triangle created by two radii and a 90° angle.

$$\frac{SectorArea}{CircleArea}=\frac{SectorAngle}{360°}=\frac{90°}{360°}=\frac{1}{4}$$

Sector OPR = $$\frac{1}{4}$$ of circle

I. The length of arc PQR? YES

Arc PQR is $$\frac{1}{4}$$ of the circumference, so
Length of arc PQR * 4 = circumference = 2πr

If PQR had arc length $$2π$$, e.g.:
(Arc length) * 4 = Circumference
$$2π * 4 = 2πr$$
$$8π = 2πr$$
$$8 = 2r$$
$$r=4$$

II. The perimeter of ∆ OPR? YES

∆ OPR is right isosceles, with
--side lengths in ratio $$r : r : r\sqrt{2}$$
Perimeter of ∆ OPR =
$$(r + r + r\sqrt{2})= (2r + r\sqrt{2})$$

If perimeter were $$8 + 4\sqrt{2}$$:
$$8 = 2r$$, and
$$r = 4$$

(Third side ratio holds: PR = $$4\sqrt{2}$$)

I would stop here. The only choice that has both I and II is answer E.

III. The length of the chord PR? YES

Chord PR, per side length ratio of 45-45-90 (right isosceles) triangles, means

Length of chord PR = $$r\sqrt{2}$$
If PR length were, e.g., 8:
$$8 = r\sqrt{2}$$
$$\frac{8}{\sqrt{2}} = r$$
$$r =(\frac{8}{\sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}})=4\sqrt{2}=r$$

All three are sufficient to find radius.

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Re: The circle in the figure above has center O. Which of the following [#permalink]

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27 Nov 2017, 11:51
1
KUDOS
Expert's post
Bunuel wrote:

The circle in the figure above has center O. Which of the following measures for the figure would be sufficient by itself to determine the radius of the circle?

I. The length of arc PQR
II. The perimeter of ∆ OPR
III. The length of the chord PR

(A) None
(B) I only
(C) II only
(D) III only
(E) I, II and III

[Reveal] Spoiler:
Attachment:
2017-11-21_1003_001.png

I. The length of arc PQR

Since we know that arc PQR corresponds to 1/4 of the circumference of the circle, we could use the information about the arc to determine the circumference and thus determine the radius.

II. The perimeter of ∆ OPR

Since we know that we have a 45-45-90 right triangle and since PO and RO are radii of the circle, then we could use the information about the perimeter to determine all sides of the triangle and thus the radius of the circle.

III. The length of the chord PR

Since chord PR is the hypotenuse of the 45-45-90 triangle, we can use that information to then determine the legs of the triangle, which also represent the radii of the circle.

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Re: The circle in the figure above has center O. Which of the following   [#permalink] 27 Nov 2017, 11:51
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