Basic properties of circles ：練習問題

A chord of a circle has a length of \( 24 \text{ cm} \). If the radius of the circle is \( 13 \text{ cm} \), what is the perpendicular distance from the centre of the circle to the chord?

In the circle with centre \(O\), \(AB\) is a diameter. \(C\) and \(D\) are points on the circle such that \(A\), \(C\), \(D\) and \(B\) lie on the circle in that order. If \(\angle BDC = 35^{\circ}\) and \(\angle CAD = 20^{\circ}\), find \(\angle ACD\).

In the figure, $A, B, C, D$ are four distinct points lying on a circle. $TD$ is a straight line tangent to the circle at $D$. It is given that the chord $AD$ is parallel to the chord $BC$ (i.e., $AD // BC$). If $\angle ADT = 70^{\circ}$, find the measure of $\angle DAC$.

In a circle with a radius of \( 25 \text{ cm} \), a chord is drawn such that its length is \( 48 \text{ cm} \). What is the perpendicular distance from the centre to the chord?

In a circle with center \(O\), \(PA\) and \(PB\) are tangents to the circle from an external point \(P\), touching the circle at points \(A\) and \(B\) respectively. If \(\angle APB = 70^\circ\), find \(\angle AOB\).

In the figure, ABCD is a cyclic quadrilateral. The side DC is extended to a point P such that D, C, and P are collinear. If \(\angle BCP = 75^\circ\) and the length of arc AD is twice the length of arc AB, find \(\angle ADB\).

In the figure, TA is tangent to the circle at A, and TBC is a straight line cutting the circle at B and C. AD is a chord parallel to the secant line TBC. If \(\angle ATB = 25^{\circ}\) and \(\angle DAC = 40^{\circ}\), calculate \(\angle ADC\).

In the figure, A, B, C, D are points on a circle, forming a cyclic quadrilateral ABCD. AT is the tangent to the circle at A. If \(\angle DAT = 70^\circ\) and \(\angle BCD = 105^\circ\), find \(\angle ADB\).

In a circle with centre \(O\), \(AB\) is a chord. \(M\) is the midpoint of \(AB\).

(a) Describe the relationship between the line segment \(OM\) and the chord \(AB\).

(b) If the radius of the circle is \(10\text{ cm}\) and the length of the chord \(AB\) is \(16\text{ cm}\), find the length of \(OM\).

In the figure, \( AB \) is a diameter of a circle with centre \( O \). \( C \) is a point on the circumference. The tangent to the circle at \( C \) intersects the extension of the diameter \( AB \) at the point \( P \). \( D \) is a point on \( AB \) such that \( CD \perp AB \).
(a) Prove that \( \triangle PAC \sim \triangle PCB \).
(b) Suppose \( AD = 9 \) and \( BD = 4 \).
    (i) Find the length of \( CD \).
    (ii) Find the length of \( PC \).
(c) A student claims that the area of \( \triangle OCP \) is more than five times the area of \( \triangle BCD \). Do you agree? Explain your answer.