Properties of circles：练习题

In the diagram, \(PA\) and \(PB\) are tangents from an external point \(P\) to a circle with centre \(O\). The points \(A\) and \(B\) lie on the circumference. If \(\angle APB = 44^\circ\), find the size of \(\angle AOB\).

In the diagram, the points \(A\), \(B\), and \(C\) lie on the circumference of a circle with centre \(O\). Given that \(\angle ABC = 110^\circ\), find the size of \(\angle OAC\).

In the diagram, \(P, Q, R,\) and \(S\) are points on a circle. \(PQ\) is parallel to \(SR\). If \(\angle PQR = 72^\circ\), find \(\angle RQS\) given that \(QS\) bisects \(\angle PQR\).

In a circle with center \( O \), \( AB \) is a diameter. Point \( C \) lies on the circumference of the circle. If \( \angle ABC = 40^\circ \), find \( \angle BAC \).

Points \(A\), \(B\), and \(C\) lie on the circumference of a circle with centre \(O\). The chord \(AC\) has a length of \(10\text{ cm}\). The perpendicular distance from the centre \(O\) to the chord \(AC\) is \(12\text{ cm}\). Calculate the radius of the circle.

In the diagram, the points \(A\), \(B\), and \(C\) lie on the circumference of a circle with centre \(O\). The straight lines \(PA\) and \(PB\) are tangents to the circle. Given that \(\angle APB = 56^\circ\), calculate the size of \(\angle ACB\).

Points \(A\), \(B\), and \(C\) lie on the circumference of a circle with centre \(O\). If \(AB\) is a diameter of the circle, \(AC = 8 \text{ cm}\), and the area of the circle is \(25\pi \text{ cm}^2\), find the length of the chord \(BC\).

Points \(A\), \(B\), \(C\), and \(D\) lie on the circumference of a circle. The chords \(AC\) and \(BD\) intersect at the point \(E\). If \(\angle CAD = 34^\circ\) and \(\angle CBD = 34^\circ\), explain why arc \(CD\) subtends equal angles at \(A\) and \(B\), and find \(\angle ACD\) if \(\angle ABD = 58^\circ\).

In the diagram, \( O \) is the centre of a circle. The line \( PQR \) is a tangent to the circle at \( Q \). The points \( S, T, \) and \( U \) lie on the circumference such that \( SU \) is a diameter and \( ST \) is parallel to the tangent \( PR \).
(a) If \( \angle SQU = 35^∘ \), calculate \( \angle QSU \), giving a reason for your answer.
(b) Prove that triangle \( OQS \) is an equilateral triangle if \( \angle SOQ = 60^∘ \).
(c) Calculate \( \angle TQR \) given that \( \angle SQU = 35^∘ \).