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

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
From circumference, find radius

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
--legs that are radii and
--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.

Answer E

Let’s analyze each answer choice:

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.

Answer: E