Length of Line Segment - help required

[rjanardhanan]

The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is (4/3)*3.14 , what is the length of line segment RU?

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

[pemdas]

arc RTU=4pi/3, find chord RU?
circumference 2pi*r=2*4*pi=8pi is corresponding to 360 degrees
hence 4pi/3 x degrees

8pi --> 360
4pi/3 --> x
x=4*120/8=60

chord RU and two radii formed by joining R and U with the center of circle form triangle, which is equilateral triangle, as the central angle is 60 and two sides (radii) are equal (60-60-60). Hence length of RU is 4

d

The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is (4/3)*3.14 , what is the length of line segment RU?

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

[neelgandham]

Solution:

Length of the arc = (A/360)*(2*pi*r), where A is the central angle of the arc in degrees and r is the radius of the circle.

Length of the arc RTU = (A/360)*(2*pi*4) = 4pi/3 (from the question). So, Angle A = 60 degrees.
In the triangle ORU.
Angle ROU = 60 Degrees
OR = OU = Radius of the circle = 4 units
Since OR=OU, Angle ORU = Angle OUR.

Sum of angles in a triangle = 180
Angle ORU + Angle OUR + Angle ROU = 180
2*Angle ORU + 60 = 180 (Angle ORU = Angle OUR)
Angle ORU = 60 Degrees

Angle ORU = Angle OUR = Angle ORU = 60. So Triangle ORU is an equilateral triangle of side 4. Therefore RU = 4 units

Answer D