Shape and Space 6

In the diagram, OA and OB are radii of the circle, centre O. TB is a tangent to the circle at point B. Angle OBA = 68°.

AB and CD are two chords of a circle, intersecting at point P inside the circle. AP = 3 cm, PB = 6 cm, CP = 4 cm.

State the alternate segment theorem and give an example of how it can be used to find an unknown angle in a circle.

A tangent touches a circle at point T. The chord AB passes through T. Given that angle ATB = 37°, write down the size of the angle in the alternate segment which is subtended by AB, and state which theorem you used.

In the diagram, O is the centre of the circle. QR and ST are two chords which intersect at right angles at point P inside the circle. If OP = 3 cm, QR = 6 cm, and ST = 8 cm, find the radius of the circle.

Explain how the alternate segment theorem can be used to find an unknown angle in a circle. Give a clear step-by-step example, including both the reasoning and the calculation.