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Segment HM is the diameter of circle O.

Secant AH intersects circle O at point T.

Segment A F is tangent to circle O at point F.

FA = 21, AT = 7, TM = 90, and TH = x.

None of the angle measures are given.

1. Solve for x. Show your work.

2. What is the measure of angle H T M? Explain how you know.

3. Based on your answers to 1 and 2, what is the radius of circle O? Show your work.

IT IS CONVENTIONAL to let the letter s symbolize the length of an arc, which is called arc length. We say in geometry that an arc "subtends" an angle θ; literally, "stretches under."

Now the circumference of a circle is an arc length. And the ratio of the circumference to the diameter is the basis of radian measure. That ratio is the definition of π.

π = C

D .

Since D = 2r, then

π = C

2r

or,

C

r = 2π .

arc lengthThat ratio -- 2π -- of the circumference of a circle to the radius, is called the radian measure of 1 revolution, which are four right angles at the center. The circumference subtends those four right angles.

arc length

Radian measure of θ = s

r

Thus the radian measure is based on ratios -- numbers -- that are actually found in the circle. The radian measure is a real number that indicates the ratio of a curved line to a straight, of an arc to the radius. For, the ratio of s to r does determine a unique central angle θ.