Shape and Space 6

In the diagram, O is the centre of the circle. The radius of the circle is 5 cm. AB and AC are chords of the circle and angle BAC = 32°.

Explain, with reasons, why the points A, B, C, and D in a circle are called concyclic points. Give a geometrical property relating their angles.

In the diagram, O is the centre of the circle. AB is a tangent to the circle at A. OC is a diameter. OA and OB are radii, and angle OAB = 28°.

ABCD is a cyclic quadrilateral. Angle ABC = 76°, angle CDA = 88°. Work out the size of angle BAD.

In a circle, the chord AB is 8 cm long and is perpendicular to the radius at its midpoint. If the radius of the circle is 5 cm, work out the perpendicular distance from the centre of the circle to the chord, giving your answer to 2 decimal places.

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.