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

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hazelnut01: Master | Next Rank: 500 Posts; Posts: 157; Joined: Sat Nov 19, 2016 5:34 amFollowed by: 4 members; Thanked: 2 times

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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Source: Beat The GMAT — Problem Solving | Wed May 03, 2017 4:24 pm

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

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

=> âˆ 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

-Jay
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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

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

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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