Question

Ray PA is tangent to circle O at point A, PA= 12 units, and the measure of AR is 60°. Answer the following and justify your answer using angle, arc, and special segments relationships with circles,

a) Find the length of the radius of circle O.
b) Find the length of line OP

Answer 1

a) radius = 6.9288 units

b) OP = 13.857 units

Explanation:

a)

As PA is tangent to the circle, the angle OAP is 90°, and if the angle of the arc AR is 60°, the angle AOP is also 60°.

So we can find the angle APO with the sum of internal angles of the triangle AOP:

90 + 60 + APO = 180

APO = 30°

Now, we can find the radius AO using the tangent relation of the angle APO, where the opposite side is the radius AO and the adjacent side is AP:

tangent(30) = AO / AP

0.5774 = AO / 12

AO = 0.5774 * 12 = 6.9288 units

b)

To find the length of OP, we can use the Pythagoras' theorem in the triangle AOP:

OP^2 = AO^2 + AP^2

OP^2 = 6.9288^2 + 12^2 = 192.01

OP = 13.857 units