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help with a geometry question

by trustkip » Fri Oct 25, 2013 3:23 pm
Please help me with this geometry question:

The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is 4*pi/3, what is the length of line segment RU?

A. 34
B. 38
C. 3
D. 4
E. 6
Last edited by trustkip on Sun Oct 27, 2013 5:01 pm, edited 1 time in total.
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by GMATGuruNY » Fri Oct 25, 2013 4:29 pm
Please edit your post above so that the correct answer is hidden by the spoiler function, as I have done here:
The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is 4*pi/3, what is the length of line segment RU?

A. 34
B. 38
C. 3
D. 4
E. 6

OA: D

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C = 2Ï€r = 2*Ï€*4 = 8Ï€.
(arc RTU)/(circumference) = (4Ï€/3) / (8Ï€) = 1/6.
Since arc RTU is 1/6 of the circumference, ∠ROU -- the central angle that intercepts arc RTU -- constitutes 1/6 of 360º:
∠ROU = 1/6 * 360 = 60.

In ∆ROU, RO and UO are both radii, so RO = UO.
Since RO = UO, ∠ORU = ∠RUO, implying that ∠ORU = ∠RUO= 60.
Thus, ∆ROU is an equilateral triangle with a side of 4, implying that RU = 4.

The correct answer is D.
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by [email protected] » Fri Oct 25, 2013 6:03 pm
Hi trustkip,

Mitch has provided the full math explanation for this question, so I'm going to focus on the "patterns" behind this question that can help you to save time.

1) If you draw a triangle inside a circle that has one point as the origin and the other two points on the circle, then that triangle will ALWAYS be ISOSCELES.

So, once you deduce that the central angle is 60 degrees, then the other 2 angles have to be EQUAL (because the triangle is isosceles). In this case, the two equal angles have to add up to 120 degrees. So, we end up with a 60-60-60 triangle, which makes it EQUILATERAL (so all the sides are equal). With a radius of 4, all sides equal 4.

2) The longest line that connects two points on the circle is the diameter.

In this case, the longest possible line would be 4(2) = 8. If you had to guess, then you could eliminate any answer greater than 8.

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