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MGMAT - Geometry

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by Anurag@Gurome » Fri Mar 18, 2011 12:33 am
diehard_gmat wrote:The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is 4Ï€/3, what is the length of line segment RU?

A. 34
B. 38
C. 3
D. 4
E. 6
Refer to the image below,
Image

The measure of the angle subtended by the arc RTU at the center of the circle = (4π/3)/4 = π/3 = 60 degrees

Hence, the triangle ORU is an equilateral one.
Hence, length of the line segment RU = length of the line segment OR = length of the radius = 4

The correct answer is D.
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by diehard_gmat » Fri Mar 18, 2011 1:08 am
Anurag@Gurome wrote:The measure of the angle subtended by the arc RTU at the center of the circle = (4π/3)/4 = π/3 = 60 degrees
How do you get that?
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by GMATGuruNY » Fri Mar 18, 2011 3:18 am
diehard_gmat wrote:The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is 4Ï€/3, what is the length of line segment RU?

A. 34
B. 38
C. 3
D. 4
E. 6
Circles display the following proportionality:

(Central Angle)/360 = (intercepted arc length)/circumference

Here's a drawing of the problem above:

Image

(Central angle ∠ROU)/360 = (intercepted arc RTU)/circumference.

Circumference = 2�r = 8�.
Arc RTU/Circumference = (4�/3)/8� = 1/6.
Thus, central angle ∠ROU = 1/6 * 360 = 60.
Since OR and OU are radii, they are equal. Thus, the angles opposite OR and OU in triangle ORU must also be equal.
Thus, ∠ORU = ∠OUR = 60.
Thus, triangle ORU is equilateral, so RU = 4.

The correct answer is D.
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