Circle theorems I：練習問題

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 \(ACB\).

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 \(PT\).

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 \(\angle ACD\).

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 \(QPR\).

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 \(\angle BAC\).

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.

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 \(\angle QPT\).

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 \(\angle BDC\).

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 \(BAP\).

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.