Tangent and sector - single tangent line

5 \includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-2_543_883_1274_630} The diagram shows a circle with centre \(O\) and radius 5 cm . The point \(P\) lies on the circle, \(P T\) is a tangent to the circle and \(P T = 12 \mathrm {~cm}\). The line \(O T\) cuts the circle at the point \(Q\).

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

7 \includegraphics[max width=\textwidth, alt={}, center]{d68c82ec-8c85-40b9-8e81-bd53c7f8dafe-3_462_956_258_593} In the diagram, \(A B\) is an arc of a circle, centre \(O\) and radius 6 cm , and angle \(A O B = \frac { 1 } { 3 } \pi\) radians. The line \(A X\) is a tangent to the circle at \(A\), and \(O B X\) is a straight line.

1. Show that the exact length of \(A X\) is \(6 \sqrt { } 3 \mathrm {~cm}\). Find, in terms of \(\pi\) and \(\sqrt { } 3\),
2. the area of the shaded region,
3. the perimeter of the shaded region.

6 \includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-3_552_734_255_703} The diagram shows a circle with radius \(r \mathrm {~cm}\) and centre \(O\). The line \(P T\) is the tangent to the circle at \(P\) and angle \(P O T = \alpha\) radians. The line \(O T\) meets the circle at \(Q\).

1. Express the perimeter of the shaded region \(P Q T\) in terms of \(r\) and \(\alpha\).
2. In the case where \(\alpha = \frac { 1 } { 3 } \pi\) and \(r = 10\), find the area of the shaded region correct to 2 significant figures.

6 \includegraphics[max width=\textwidth, alt={}, center]{5df7bd9f-31cc-41a3-b1c0-3ee9366e6d8a-10_499_922_262_607} The diagram shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The points \(A\) and \(B\) lie on the circle and \(A T\) is a tangent to the circle. Angle \(A O B = \theta\) radians and \(O B T\) is a straight line.

1. Express the area of the shaded region in terms of \(r\) and \(\theta\).
2. In the case where \(r = 3\) and \(\theta = 1.2\), find the perimeter of the shaded region.

11 \includegraphics[max width=\textwidth, alt={}, center]{2f48a6ee-e8ce-47e4-a07f-2c55a6904e7d-3_661_953_767_596} The diagram shows a circle, centre \(O\), radius \(r\). The points \(R\) and \(S\) lie on the circumference of the circle, and the line \(R T\) is a tangent to the circle at \(R\). The angle \(R O S\) is \(\theta\) radians where \(0 < \theta < \frac { 1 } { 2 } \pi\).

1. Find expressions for the perimeter, \(P\), and the area, \(A\), of the shaded region in terms of \(r\) and \(\theta\).
2. Hence show that \(A \neq r P\).

\includegraphics{figure_3} The diagram shows triangle \(ABC\) which is right-angled at \(A\). Angle \(ABC = \frac{1}{4}\pi\) radians and \(AC = 8\) cm. The points \(D\) and \(E\) lie on \(BC\) and \(BA\) respectively. The sector \(ADE\) is part of a circle with centre \(A\) and is such that \(BDC\) is the tangent to the arc \(DE\) at \(D\).

1. Find the length of \(AD\). [3]
2. Find the area of the shaded region. [3]