1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta

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 \(\sqrt { } 3\).

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.

5 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-2_385_403_1866_872} The diagram shows a circle with centre \(O\). The circle is divided into two regions, \(R _ { 1 }\) and \(R _ { 2 }\), by the radii \(O A\) and \(O B\), where angle \(A O B = \theta\) radians. The perimeter of the region \(R _ { 1 }\) is equal to the length of the major \(\operatorname { arc } A B\).

1. Show that \(\theta = \pi - 1\).
2. Given that the area of region \(R _ { 1 }\) is \(30 \mathrm {~cm} ^ { 2 }\), find the area of region \(R _ { 2 }\), correct to 3 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-3_744_675_255_735} The diagram shows a metal plate \(A B C D E F\) which has been made by removing the two shaded regions from a circle of radius 10 cm and centre \(O\). The parallel edges \(A B\) and \(E D\) are both of length 12 cm .

1. Show that angle \(D O E\) is 1.287 radians, correct to 4 significant figures.
2. Find the perimeter of the metal plate.
3. Find the area of the metal plate.

9 \includegraphics[max width=\textwidth, alt={}, center]{53839c8c-07ea-4545-9c00-a6884aa2afc3-3_387_1175_1781_486} In the diagram, \(O A B\) is an isosceles triangle with \(O A = O B\) and angle \(A O B = 2 \theta\) radians. Arc \(P S T\) has centre \(O\) and radius \(r\), and the line \(A S B\) is a tangent to the \(\operatorname { arc } P S T\) at \(S\).

1. Find the total area of the shaded regions in terms of \(r\) and \(\theta\).
2. In the case where \(\theta = \frac { 1 } { 3 } \pi\) and \(r = 6\), find the total perimeter of the shaded regions, leaving your answer in terms of \(\sqrt { } 3\) and \(\pi\).

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.

3 \includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-2_584_659_575_742} In the diagram, \(A B C\) is an equilateral triangle of side 2 cm . The mid-point of \(B C\) is \(Q\). An arc of a circle with centre \(A\) touches \(B C\) at \(Q\), and meets \(A B\) at \(P\) and \(A C\) at \(R\). Find the total area of the shaded regions, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).

3 \includegraphics[max width=\textwidth, alt={}, center]{d0074ac8-42d2-49f4-a417-4a348537bccc-2_492_682_708_733} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius 8 cm . Angle \(B O A\) is \(\alpha\) radians. \(O A C\) is a semicircle with diameter \(O A\). The area of the semicircle \(O A C\) is twice the area of the sector \(O A B\).

1. Find \(\alpha\) in terms of \(\pi\).
2. Find the perimeter of the complete figure in terms of \(\pi\).

4 \includegraphics[max width=\textwidth, alt={}, center]{fe4c3555-5736-48c4-b61a-9f6b9a1ee46e-2_645_652_1023_744} The diagram shows a square \(A B C D\) of side 10 cm . The mid-point of \(A D\) is \(O\) and \(B X C\) is an arc of a circle with centre \(O\).

1. Show that angle \(B O C\) is 0.9273 radians, correct to 4 decimal places.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

2 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-2_501_641_461_753} The diagram shows a circle \(C\) with centre \(O\) and radius 3 cm . The radii \(O P\) and \(O Q\) are extended to \(S\) and \(R\) respectively so that \(O R S\) is a sector of a circle with centre \(O\). Given that \(P S = 6 \mathrm {~cm}\) and that the area of the shaded region is equal to the area of circle \(C\),

1. show that angle \(P O Q = \frac { 1 } { 4 } \pi\) radians,
2. find the perimeter of the shaded region.

6 \includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-3_625_897_260_623} The diagram shows triangle \(A B C\) in which \(A B\) is perpendicular to \(B C\). The length of \(A B\) is 4 cm and angle \(C A B\) is \(\alpha\) radians. The arc \(D E\) with centre \(A\) and radius 2 cm meets \(A C\) at \(D\) and \(A B\) at \(E\). Find, in terms of \(\alpha\),

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

4 \includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-2_358_618_1082_762} The diagram shows a sector of a circle with radius \(r \mathrm {~cm}\) and centre \(O\). The chord \(A B\) divides the sector into a triangle \(A O B\) and a segment \(A X B\). Angle \(A O B\) is \(\theta\) radians.

1. In the case where the areas of the triangle \(A O B\) and the segment \(A X B\) are equal, find the value of the constant \(p\) for which \(\theta = p \sin \theta\).
2. In the case where \(r = 8\) and \(\theta = 2.4\), find the perimeter of the segment \(A X B\).

3 \includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-2_485_623_790_760} The diagram shows part of a circle with centre \(O\) and radius 6 cm . The chord \(A B\) is such that angle \(A O B = 2.2\) radians. Calculate

1. the perimeter of the shaded region,
2. the ratio of the area of the shaded region to the area of the triangle \(A O B\), giving your answer in the form \(k : 1\).

11 \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-4_519_560_260_797} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). The point \(C\) on \(O B\) is such that angle \(A C O\) is a right angle. Angle \(A O B\) is \(\alpha\) radians and is such that \(A C\) divides the sector into two regions of equal area.

1. Show that \(\sin \alpha \cos \alpha = \frac { 1 } { 2 } \alpha\). It is given that the solution of the equation in part (i) is \(\alpha = 0.9477\), correct to 4 decimal places.
2. Find the ratio perimeter of region \(O A C\) : perimeter of region \(A C B\), giving your answer in the form \(k : 1\), where \(k\) is given correct to 1 decimal place.
3. Find angle \(A O B\) in degrees. {www.cie.org.uk} after the live examination series. }

7 \includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-3_408_451_721_845} In the diagram, \(A O B\) is a quarter circle with centre \(O\) and radius \(r\). The point \(C\) lies on the arc \(A B\) and the point \(D\) lies on \(O B\). The line \(C D\) is parallel to \(A O\) and angle \(A O C = \theta\) radians.

1. Express the perimeter of the shaded region in terms of \(r , \theta\) and \(\pi\).
2. For the case where \(r = 5 \mathrm {~cm}\) and \(\theta = 0.6\), find the area 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.

8 \includegraphics[max width=\textwidth, alt={}, center]{028c7979-6b24-42d0-9857-c616a169b2b2-14_590_691_260_726} In the diagram, \(O A X B\) is a sector of a circle with centre \(O\) and radius 10 cm . The length of the chord \(A B\) is 12 cm . The line \(O X\) passes through \(M\), the mid-point of \(A B\), and \(O X\) is perpendicular to \(A B\). The shaded region is bounded by the chord \(A B\) and by the arc of a circle with centre \(X\) and radius \(X A\).

1. Show that angle \(A X B\) is 2.498 radians, correct to 3 decimal places.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

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.

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.

5 \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-06_323_775_260_685} The diagram shows a triangle \(O A B\) in which angle \(O A B = 90 ^ { \circ }\) and \(O A = 5 \mathrm {~cm}\). The arc \(A C\) is part of a circle with centre \(O\). The arc has length 6 cm and it meets \(O B\) at \(C\). Find the area of the shaded region.

3 A sector of a circle of radius \(r \mathrm {~cm}\) has an area of \(A \mathrm {~cm} ^ { 2 }\). Express the perimeter of the sector in terms of \(r\) and \(A\).

5 \includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-06_355_634_255_753} The diagram shows a semicircle with diameter \(A B\), centre \(O\) and radius \(r\). The point \(C\) lies on the circumference and angle \(A O C = \theta\) radians. The perimeter of sector \(B O C\) is twice the perimeter of sector \(A O C\). Find the value of \(\theta\) correct to 2 significant figures.

9

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_433_476_264_872} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} In Fig. 1, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). \(A X\) is the tangent at \(A\) to the arc \(A B\) and angle \(B A X = \alpha\).
   1. Show that angle \(A O B = 2 \alpha\).
   2. Find the area of the shaded segment in terms of \(r\) and \(\alpha\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_451_503_1162_861} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} In Fig. 2, \(A B C\) is an equilateral triangle of side 4 cm . The lines \(A X , B X\) and \(C X\) are tangents to the equal circular \(\operatorname { arcs } A B , B C\) and \(C A\). Use the results in part (a) to find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
   [0pt] [6]

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.

6 \includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-3_293_502_269_826} The diagram shows the sector \(O P Q\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The angle \(P O Q\) is \(\theta\) radians and the perimeter of the sector is 20 cm .

1. Show that \(\theta = \frac { 20 } { r } - 2\).
2. Hence express the area of the sector in terms of \(r\).
3. In the case where \(r = 8\), find the length of the chord \(P Q\).