a central angle intercepts an arc on a circle equal in length to the diameter of the circle. find the measure, in radians, of the central angle.

d = r theta = (d/2)theta

theta = 2 radians

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To find the measure of the central angle in radians, we need to use the given information: "a central angle intercepts an arc on a circle equal in length to the diameter of the circle."

We know that the length of the arc intercepted by the central angle is equal to the diameter of the circle. The diameter of a circle is twice the radius. Let's denote the radius of the circle as 'r.'

Therefore, the length of the arc intercepted by the central angle is 2r.

The circumference of a circle is given by the formula C = 2πr, where 'C' represents the circumference and 'π' is a mathematical constant approximately equal to 3.14159.

Since the arc intercepted by the central angle is equal to the diameter, it is also equal to half the circumference of the circle, which is given by C/2.

Setting C/2 equal to 2r, we have:

C/2 = 2r

Since C = 2πr, we can substitute it into the equation:

(2πr)/2 = 2r

Simplifying, we cancel out the 2's:

πr = 2r

Now, we can divide both sides of the equation by 'r' to solve for π:

π = 2

Therefore, the measure of the central angle in radians is 2 radians.