Tangent and sector - two tangents from external point

7 \includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-3_545_759_269_694} The diagram shows a circle with centre \(O\) and radius 8 cm . Points \(A\) and \(B\) lie on the circle. The tangents at \(A\) and \(B\) meet at the point \(T\), and \(A T = B T = 15 \mathrm {~cm}\).

1. Show that angle \(A O B\) is 2.16 radians, correct to 3 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

5 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-2_586_682_1726_733} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius 12 cm . The lines \(A X\) and \(B X\) are tangents to the circle at \(A\) and \(B\) respectively. Angle \(A O B = \frac { 1 } { 3 } \pi\) radians.

1. Find the exact length of \(A X\), giving your answer in terms of \(\sqrt { } 3\).
2. Find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).

6 \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-08_454_684_255_726} The diagram shows points \(A\) and \(B\) on a circle with centre \(O\) and radius \(r\). The tangents to the circle at \(A\) and \(B\) meet at \(T\). The shaded region is bounded by the minor \(\operatorname { arc } A B\) and the lines \(A T\) and \(B T\). Angle \(A O B\) is \(2 \theta\) radians.

1. In the case where the area of the sector \(A O B\) is the same as the area of the shaded region, show that \(\tan \theta = 2 \theta\).
2. In the case where \(r = 8 \mathrm {~cm}\) and the length of the minor \(\operatorname { arc } A B\) is 19.2 cm , find the area of the shaded region.

3 \includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-2_330_634_753_758} In the diagram, \(O P Q\) is a sector of a circle, centre \(O\) and radius \(r \mathrm {~cm}\). Angle \(Q O P = \theta\) radians. The tangent to the circle at \(Q\) meets \(O P\) extended at \(R\).

1. Show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the shaded region is given by \(A = \frac { 1 } { 2 } r ^ { 2 } ( \tan \theta - \theta )\).
2. In the case where \(\theta = 0.8\) and \(r = 15\), evaluate the length of the perimeter of the shaded region.

8 \includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-12_560_574_258_781} The diagram shows a sector \(O A C\) of a circle with centre \(O\). Tangents \(A B\) and \(C B\) to the circle meet at \(B\). The arc \(A C\) is of length 6 cm and angle \(A O C = \frac { 3 } { 8 } \pi\) radians.

1. Find the length of \(O A\) correct to 4 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

4 \includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-06_517_768_262_685} The diagram shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). Points \(A\) and \(B\) lie on the circle and angle \(A O B = 2 \theta\) radians. The tangents to the circle at \(A\) and \(B\) meet at \(T\).

1. Express the perimeter of the shaded region in terms of \(r\) and \(\theta\).
2. In the case where \(r = 5\) and \(\theta = 1.2\), find the area of the shaded region.

10 Arrowline Enterprises is considering two possible logos: \begin{figure}[h] \end{figure} Fig. 10.2

1. Fig. 10.1 shows the first logo ABCD . It is symmetrical about AC . Find the length of AB and hence find the area of this logo.
2. Fig. 10.2 shows a circle with centre O and radius 12.6 cm . ST and RT are tangents to the circle and angle SOR is 1.82 radians. The shaded region shows the second logo. Show that \(\mathrm { ST } = 16.2 \mathrm {~cm}\) to 3 significant figures.
   Find the area and perimeter of this logo.

5 The diagram shows a sector \(O P Q\) of a circle with centre \(O\). \includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-3_305_531_1105_758} The radius of the circle is 18 m and the angle \(P O Q\) is \(\frac { 2 \pi } { 3 }\) radians.

1. Find the length of the arc \(P Q\), giving your answer as a multiple of \(\pi\).
2. The tangents to the circle at the points \(P\) and \(Q\) meet at the point \(T\), and the angles \(T P O\) and \(T Q O\) are both right angles, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-3_597_529_1848_758}
   1. Angle \(P T Q = \alpha\) radians. Find \(\alpha\) in terms of \(\pi\).
   2. Find the area of the shaded region bounded by the \(\operatorname { arc } P Q\) and the tangents \(T P\) and \(T Q\), giving your answer to three significant figures.

\includegraphics{figure_8} In the diagram, \(AB\) is an arc of a circle with centre \(O\) and radius \(r\). The line \(XB\) is a tangent to the circle at \(B\) and \(A\) is the mid-point of \(OX\).

1. Show that angle \(AOB = \frac{1}{3}\pi\) radians. [2]

Express each of the following in terms of \(r\), \(\pi\) and \(\sqrt{3}\):

1. the perimeter of the shaded region, [3]
2. the area of the shaded region. [2]

\includegraphics{figure_2} In the diagram, \(OADC\) is a sector of a circle with centre \(O\) and radius 3 cm. \(AB\) and \(CB\) are tangents to the circle and angle \(ABC = \frac{1}{4}\pi\) radians. Find, giving your answer in terms of \(\sqrt{3}\) and \(\pi\),

1. the perimeter of the shaded region, [3]
2. the area of the shaded region. [3]

\includegraphics{figure_6} The diagram shows points \(A\) and \(B\) on a circle with centre \(O\) and radius \(r\). The tangents to the circle at \(A\) and \(B\) meet at \(T\). The shaded region is bounded by the minor arc \(AB\) and the lines \(AT\) and \(BT\). Angle \(AOB\) is \(2\theta\) radians.

1. In the case where the area of the sector \(AOB\) is the same as the area of the shaded region, show that \(\tan \theta = 2\theta\). [3]
2. In the case where \(r = 8\) cm and the length of the minor arc \(AB\) is 19.2 cm, find the area of the shaded region. [3]