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The points R, T, and U lie on a circle that has radius 4. If

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by hazelnut01 » Wed May 03, 2017 4:24 pm
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. 4/3
B. 8/3
C. 3
D. 4
E. 6
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by Jay@ManhattanReview » Wed May 03, 2017 9:35 pm
ziyuenlau 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. 4/3
B. 8/3
C. 3
D. 4
E. 6
Circumference of the circle = 2Ï€r = 2Ï€*4 = 8Ï€;

Length of the arc = 4Ï€/3

Thus, the ratio of 'Length of the arc' to 'Circumference' = (4Ï€/3)/8Ï€ = 1/6

=> ∠RCU = (1/6)*360 = 60 deg.

Thus, RCU is an equilateral triangle.

=> RU = radius = 4

The correct answer: D

Hope this helps!

Relevant book: Manhattan Review GMAT Geometry Guide

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by Jeff@TargetTestPrep » Wed Nov 22, 2017 12:19 pm
hazelnut01 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. 4/3
B. 8/3
C. 3
D. 4
E. 6
We can let the center of the circle be C. Since the radius of circle C is 4, its circumference is 2Ï€r = 2Ï€(4) = 8Ï€.

We can use the following proportion to determine the central angle that intercepts arc RTU:

x/360 = (4Ï€/3)/8Ï€

x/360 = 4Ï€/24Ï€

x/360 = 1/6

x = 60

Now triangle RUC (i.e., the triangle formed by radii RC and UC and chord RU) is at least an isosceles triangle, since RC = UC = 4. However, since x = 60 degrees, or angle RCU = 60 degrees, triangle RUC must be equilateral because angles RUC and URC are also 60 degrees. Since triangle RUC is equilateral, RU = RC = 4.

Answer: D

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