crackgmat007 wrote:The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is (4pi)/3 , what is the length of line segment RU?

A. 4/3
B. 8/3
C. 3
D. 4
E. 6

Need help. I was able to work till the angle measure 60. I was trying to conclude that 60 means equilateral triangle, but was not able to visualize. Can someone solve this pls? Tx.

formula for arc length is (x/360)*2pi(r). They give us arc length and radius. so plug in to formula and get 60 as you did.

Now here's the trick part. I am going to label the center of my circle O. Because both 2 sides of the triangle formed ROU equal 4 and are equal, we have an iscosceles situation. So set up equation to solve other two angles. 60+2x=180 --> x=60. Which means we have an equilateral triangle ---> Therefore RU = 4.

You got this!

R,T and U are lie on the circle. The points might be something like the one I attached.

Circumference of the circle with radius 4 = 2 *pi* 4 = 8*pi

if 8*pi is 360 degree, then 4*pi/3 is, 60 degree.

We found the angle.

Case 1:

Take R, T and U lie on the circle (Like the one I attached).

You don't need to find anything here. Its already there. 4 *pi/3; Assume pi = 3; The answer is 4.

Case 2:

Take any point as the center of the circle. The triangle will be a equilateral triangle. And the answer is 4 again.

Thanks guys.

@UMA

Why we need to find a Angle here? Given that Length of RTU is 4(pi)/3. Length of RU ~ 4.

What is the purpose of finding angle? Are we trying to prove that T is between R and U.

How to confirm the formed triangle is Isosceles? 2 Sides are equal. What about two angles? Is there any formula related to this?

Please help...

regards,
hhk

hariharakarthi wrote:@UMA

Why we need to find a Angle here? Given that Length of RTU is 4(pi)/3. Length of RU ~ 4.

What is the purpose of finding angle? Are we trying to prove that T is between R and U.

How to confirm the formed triangle is Isosceles? 2 Sides are equal. What about two angles? Is there any formula related to this?

Please help...

regards,
hhk

We're not assuming that the triangle is isosceles. If you check my previous post, I listed two cases.

We know the radius. its 4. With that, we can find the circumference of the circle.

2*pi*4 = 8*pi

The total angle of the circle is 360. We have to find the angle for 4*pi/3

8*pi/360 = (4*pi/3)/x

x = 60.

The angle of RTU is 60 and center from R and U is 4 (radius). So, its an isosceles triangle. therefore, RU is also 4.

Hope this helps.