Shape and Space 6

Chords AP and BP intersect inside a circle at point P. If AP = 4 cm, BP = 6 cm, and CP = 8 cm, find DP.

In a circle, points A, B, C, D lie on the circumference. Angle ABC = 80°. Find angle CDA. Give a reason for your answer.

In the circle below, O is the centre. The distance from O to the chord AB is 5 cm. The radius of the circle is 13 cm. Find the length of the chord AB.

Name the theorem that states, “The angle in a semicircle is a right angle.”

Which of the following statements is always true for any cyclic quadrilateral?

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.