The chord ST of a circle is equal to the radius, r, of the circle. Find the length of arc ST.
A. πr/6
B. πr/2
C. πr/12
D. πr/3
The length of an arc is given by the formula s = θr, where θ is the angle (in radians) subtended by the arc at the center of the circle and r is the radius of the circle.
In this case, since chord ST is equal to the radius r, we can consider the triangle STR formed by the chord and two radii.
Since a triangle in a circle with one side equal to the radius is an equilateral triangle, all angles of triangle STR are equal to 60 degrees (or π/3 radians).
Thus, the length of arc ST is s = θr = (π/3)r.
Therefore, the answer is D. πr/3.
Since chord ST is equal to the radius r of the circle, we can infer that angle SOT is 60 degrees (since it subtends an arc of 120 degrees, and is half of that).
To find the length of arc ST, we need to know the circumference of the circle. The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius.
Since the central angle SOT is 60 degrees, the arc ST is 1/6th of the circumference (since 60 degrees is 1/6th of 360 degrees, which is the full circle).
Therefore, the length of arc ST is (1/6) * C, where C is the circumference.
Substituting the formula for the circumference into the equation, we get:
Length of arc ST = (1/6) * 2πr
Simplifying further, we have:
Length of arc ST = (1/3) πr
Therefore, the correct answer is:
D. πr/3