角、多邊形與幾何推理：練習題

Points \(A\), \(B\), \(C\), and \(D\) lie on a circle. The lines \(AB\) and \(DC\) are produced to meet at a point \(P\) outside the circle. If \(PB = 6\text{ cm}\), \(AB = 4\text{ cm}\), and \(PD = 15\text{ cm}\), find the length of \(CD\).

A regular polygon has an interior angle that is five times the size of its exterior angle. Find the number of sides of this polygon.

In a circle with center \(O\), \(PA\) and \(PB\) are tangents to the circle from an external point \(P\), where \(A\) and \(B\) are points on the circumference. A third tangent is drawn at a point \(C\) on the minor arc \(AB\), intersecting \(PA\) at \(X\) and \(PB\) at \(Y\). If the perimeter of triangle \(PXY\) is \(24\) cm, find the length of the tangent \(PA\).

In a circle with centre \(O\), \(AC\) is a diameter. \(B\) is a point on the circumference such that angle \(BAC = 35^{\circ}\). A tangent to the circle at \(B\) meets the line \(AC\) produced at point \(D\). Calculate the size of angle \(BDC\).

The interior angle of a regular polygon is \(162^\circ\). Calculate the number of sides of this polygon.

The diagram shows a regular polygon where each interior angle is \(140^\circ\).

(a) Calculate the size of one exterior angle of this polygon.
(b) Calculate the number of sides, \(n\), of this polygon.
(c) Find the sum of the interior angles of a regular polygon that has \(n + 2\) sides.

In the circle with center \(O\), \(ABCD\) is a cyclic quadrilateral. The length of the chord \(AB = 10\) cm and \(BC = 24\) cm. Given that \(\angle ABC = 90^\circ\):

(a) Explain why \(AC\) must be a diameter of the circle.
(b) Calculate the area of the circle, leaving your answer in terms of \(\pi\).
(c) If \(CD = DA\), calculate the area of the quadrilateral \(ABCD\) to 1 decimal place.
(d) Find the size of \(\angle DAC\) in degrees.