Circle theorems II: แบบฝึกหัด

Tangents PA and PB touch a circle at A and B from an external point P. If angle APB = \(64^\circ\), find the size of angle APO, where O is the centre of the circle.

Two equal chords PQ and RS are located in a circle with radius 10 cm. If the chords are both 6 cm from the centre O, find the length of each chord.

In a circle with centre \(O\), tangents are drawn from an external point \(P\) to touch the circle at points \(A\) and \(B\). If the length of the chord \(AB\) is equal to the radius of the circle, calculate the size of \(\angle APB\).

Two tangents PA and PB are drawn from an external point P to a circle. If the length of PA is \(15\text{ cm}\), what is the length of PB?

In a circle, a chord of length \(24\text{ cm}\) is at a distance of \(x\text{ cm}\) from the centre. If the radius of the circle is \((x + 4)\text{ cm}\), find the value of \(x\).

In a circle with centre \(O\), two tangents are drawn from an external point \(P\) to touch the circle at points \(A\) and \(B\). If the distance from \(P\) to \(A\) is \(12\text{ cm}\), find the length of the line segment \(PB\).

Two tangents PA and PB are drawn from an external point P to a circle with centre O. If angle APB is \(70^\circ\), calculate the angle AOB. State the geometrical reasons for your steps.

In the diagram, \(TA\) and \(TB\) are tangents to the circle with centre \(O\) at points \(A\) and \(B\) respectively. The radius of the circle is \(5 \text{ cm}\) and \(OT = 13 \text{ cm}\).
(a) Calculate the length of the tangent \(TA\).

(b) Find angle \(AOB\). Give your answer correct to one decimal place.

(c) Calculate the area of the quadrilateral \(TAOB\).

(d) Calculate the area of the minor sector \(AOB\). Give your answer correct to 3 significant figures.

In a circle with centre \( O \), two chords \( AB \) and \( PQ \) are known to be equal in length. The perpendicular distance from \( O \) to the chord \( AB \) is given by the expression \( 3h - 5 \), and the perpendicular distance from \( O \) to the chord \( PQ \) is \( h + 7 \).

(a) Calculate the value of \( h \). Give a geometrical reason for your method.

(b) Tangents are drawn from an external point \( T \) to touch the circle at points \( A \) and \( B \). The distance \( TA \) is \( 15 \text{ cm} \). State the length of \( TB \) and give a reason for your answer.

(c) The radius of the circle is \( 17 \text{ cm} \).
(i) Calculate the length of the chord \( AB \).
(ii) Calculate the distance from the external point \( T \) to the centre \( O \).