Cambridge IGCSE · Mathematics (0580)

Circle theorems I: Practice Questions

5 multiple-choice questions marked as you go, and 5 written questions with worked solutions. All on Circle theorems I.

10 questions25 marksFree, no account

Question 1

1 mark

In the diagram, points \(A\) and \(B\) lie on a circle with centre \(O\). Point \(C\) lies on the major arc \(AB\). If the minor angle \(AOB = 130^\circ\), find the size of angle \(ACB\).

Question 2

1 mark

In a circle with centre \(O\), the point \(P\) lies on the circumference. The line \(PT\) is a tangent to the circle at point \(P\). If the radius of the circle is \(9\text{ cm}\) and the length of the line segment from \(O\) to a point \(T\) on the tangent is \(15\text{ cm}\), calculate the length of \(PT\).

Question 3

1 mark

In the diagram, \(ABCD\) is a cyclic quadrilateral. The side \(AB\) is produced to \(E\). If \(\angle CBE = 82^\circ\) and \(\angle CAD = 35^\circ\), calculate the size of \(\angle ACD\).

Question 4

1 mark

In the diagram, \(PQ\) is a diameter of a circle with centre \(O\). The point \(R\) lies on the circumference. If angle \(PQR = 33^\circ\), find the size of angle \(QPR\).

Question 5

1 mark

In the diagram, points \(A\), \(B\), and \(C\) lie on the circumference of a circle with centre \(O\). If \(AC\) is a diameter and \(\angle BOC = 70^\circ\), find the size of \(\angle BAC\).

Question 6

2 marks

A circle has a diameter AB and a point C on its circumference. If the angle ABC is \(35^\circ\), calculate the size of angle BAC.

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Question 7

3 marks

Points \(P, Q,\) and \(R\) lie on the circumference of a circle with centre \(O\). The line \(PT\) is a tangent to the circle at point \(P\). Given that \(\angle OPT = 90^\circ\) and the radius \(OP\) makes an angle of \(38^\circ\) with the chord \(PQ\), find the size of the angle \(\angle QPT\).

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Question 8

5 marks

In the diagram, points \(A, B, C,\) and \(D\) form a cyclic quadrilateral. The side \(AB\) is produced to point \(E\). If \(\angle CBE = 82^\circ\) and \(\angle ADB = 40^\circ\), calculate the size of \(\angle BDC\).

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Question 9

4 marks

Points \(A\) and \(B\) lie on the circumference of a circle with centre \(O\). The line \(PAT\) is a tangent to the circle at point \(A\). Triangle \(OAB\) is an equilateral triangle.
(a) State the size of angle \(OAP\) and give a geometrical reason for your answer.
(b) Find the size of angle \(OAB\).
(c) Calculate the size of angle \(BAP\).

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Question 10

6 marks

In the diagram, \(A, B, C,\) and \(D\) are points on the circumference of a circle. \(AB\) is a diameter of the circle. Angle \(ACD = 32^\circ\) and angle \(CAD = 40^\circ\).
(a) Find angle \(ABD\). Give a geometrical reason for your answer.
(b) Find the size of angle \(ACB\).
(c) Calculate the size of angle \(BCD\).
(d) Find angle \(CBD\). Give a geometrical reason for your answer.

Write your answer out first, then check it against the worked solution.

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