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Good Circle Problem

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by tonebeeze » Tue Apr 26, 2011 2:59 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

OA = D
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by Tani » Tue Apr 26, 2011 3:33 pm
Draw your circle with the center at O. You have a triangle RUO.

Your arc RTU is 4pi/3. The Circumference of the circle is 8pi. Therefore, the arc is 1/6 of hte circumference so the central angel is 1/6 of 360 degrees.

Now draw a third Radius from the center perpendicular to line RU. You will have bisected the central angle and created two 30:60:90 triangles. The hypotenuses of those triangles are the radii (=4), so the sides opposite the 30 degree angles at the center are each 2 units. Therefore, the length of RU is 2*2 = 4 units.
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by Anurag@Gurome » Tue Apr 26, 2011 8:26 pm
tonebeeze 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

OA = D
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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