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

4. \begin{figure}[h] \end{figure} Not to scale Figure 2 shows a flag \(X Y W Z X\). The flag consists of a triangle \(X Y Z\) joined to a sector \(Z Y W\) of a circle with radius 5 cm and centre \(Y\). The angle of the sector, angle \(Z Y W\), is 0.7 radians. The points \(X , Y\) and \(W\) lie on a straight line with \(X Y = 7 \mathrm {~cm}\) and \(Y W = 5 \mathrm {~cm}\). Find

1. the area of the sector \(Z Y W\) in \(\mathrm { cm } ^ { 2 }\),
2. the area of the flag, in \(\mathrm { cm } ^ { 2 }\), to 2 decimal places,
3. the length of the perimeter, \(X Y W Z X\), of the flag, in cm to 2 decimal places.

5. \begin{figure}[h] \end{figure} The shaded area in Fig. 1 shows a badge \(A B C\), where \(A B\) and \(A C\) are straight lines, with \(A B = A C = 8 \mathrm {~mm}\). The curve \(B C\) is an arc of a circle, centre \(O\), where \(O B = O C =\) 8 mm and \(O\) is in the same plane as \(A B C\). The angle \(B A C\) is 0.9 radians.

1. Find the perimeter of the badge.
2. Find the area of the badge.

7 \includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-3_563_639_1379_753} The diagram shows an equilateral triangle \(A B C\) with sides of length 12 cm . The mid-point of \(B C\) is \(O\), and a circular arc with centre \(O\) joins \(D\) and \(E\), the mid-points of \(A B\) and \(A C\).

1. Find the length of the arc \(D E\), and show that the area of the sector \(O D E\) is \(6 \pi \mathrm {~cm} ^ { 2 }\).
2. Find the exact area of the shaded region.

4 \includegraphics[max width=\textwidth, alt={}, center]{58680cd3-8744-42ee-83d4-35056592b2d0-2_647_797_1323_680} The diagram shows a sector \(O A B\) of a circle with centre \(O\). The angle \(A O B\) is 1.8 radians. The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively. It is given that \(O A = O B = 20 \mathrm {~cm}\) and \(O C = O D = 15 \mathrm {~cm}\). The shaded region is bounded by the arcs \(A B\) and \(C D\) and by the lines \(C A\) and \(D B\).

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

1 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 11 cm . The angle \(A O B\) is 0.7 radians. Find the area of the segment shaded in the diagram.

2 \includegraphics[max width=\textwidth, alt={}, center]{387a37c4-0997-484c-8e28-954639169ebe-2_579_895_817_625} A sector \(O A B\) of a circle of radius \(r \mathrm {~cm}\) has angle \(\theta\) radians. The length of the arc of the sector is 12 cm and the area of the sector is \(36 \mathrm {~cm} ^ { 2 }\) (see diagram).

1. Write down two equations involving \(r\) and \(\theta\).
2. Hence show that \(r = 6\), and state the value of \(\theta\).
3. Find the area of the segment bounded by the arc \(A B\) and the chord \(A B\).

7 \includegraphics[max width=\textwidth, alt={}, center]{367db494-294e-4b53-b9e8-fd2a69fb6069-3_476_1018_1000_566} The diagram shows a triangle \(A B C\), and a sector \(A C D\) of a circle with centre \(A\). It is given that \(A B = 11 \mathrm {~cm} , B C = 8 \mathrm {~cm}\), angle \(A B C = 0.8\) radians and angle \(D A C = 1.7\) radians. The shaded segment is bounded by the line \(D C\) and the arc \(D C\).

1. Show that the length of \(A C\) is 7.90 cm , correct to 3 significant figures.
2. Find the area of the shaded segment.
3. Find the perimeter of the shaded segment.

8 \includegraphics[max width=\textwidth, alt={}, center]{e429080f-8634-46bc-b451-7b13b871e518-3_300_744_1046_703} The diagram shows a triangle \(A B C\), where angle \(B A C\) is 0.9 radians. \(B A D\) is a sector of the circle with centre A and radius AB .

1. The area of the sector \(B A D\) is \(16.2 \mathrm {~cm} ^ { 2 }\). Show that the length of \(A B\) is 6 cm .
2. The area of triangle \(A B C\) is twice the area of sector \(B A D\). Find the length of \(A C\).
3. Find the perimeter of the region \(B C D\).

7 \begin{figure}[h] \end{figure} Fig. 7 shows a sector of a circle of radius 5 cm which has angle \(\theta\) radians. The sector has area \(30 \mathrm {~cm} ^ { 2 }\).

1. Find \(\theta\).
2. Hence find the perimeter of the sector.

7 \begin{figure}[h] \end{figure} Fig. 7 shows a sector of a circle of radius 5 cm which has angle \(\theta\) radians. The sector has area \(30 \mathrm {~cm} ^ { 2 }\).

1. Find \(\theta\).
2. Hence find the perimeter of the sector.

1 Given that \(140 ^ { \circ } = k \pi\) radians, find the exact value of \(k\).

7 In Fig. 7, A and B are points on the circumference of a circle with centre O . Angle \(\mathrm { AOB } = 1.2\) radians. The arc length AB is 6 cm . \begin{figure}[h] \end{figure}

1. Calculate the radius of the circle.
2. Calculate the length of the chord AB .

11 Fig. 11.1 shows a village green which is bordered by 3 straight roads \(A B , B C\) and \(C A\). The road AC runs due North and the measurements shown are in metres. \begin{figure}[h] \end{figure}

1. Calculate the bearing of B from C , giving your answer to the nearest \(0.1 ^ { \circ }\).
2. Calculate the area of the village green. The road AB is replaced by a new road, as shown in Fig. 11.2. The village green is extended up to the new road. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b4c0b4b0-f13c-49a9-9f98-f86f28d1f577-4_440_1002_1436_737} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure} The new road is an arc of a circle with centre O and radius 130 m .
3. (A) Show that angle AOB is 1.63 radians, correct to 3 significant figures.
   (B) Show that the area of land added to the village green is \(5300 \mathrm {~m} ^ { 2 }\) correct to 2 significant figures.

11

1. The course for a yacht race is a triangle, as shown in Fig. 11.1. The yachts start at A , then travel to B , then to C and finally back to A . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-4_661_869_404_680} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
   \end{figure} (A) Calculate the total length of the course for this race.
   (B) Given that the bearing of the first stage, AB , is \(175 ^ { \circ }\), calculate the bearing of the second stage, BC.
2. Fig. 11.2 shows the course of another yacht race. The course follows the arc of a circle from P to \(Q\), then a straight line back to \(P\). The circle has radius 120 m and centre \(O\); angle \(P O Q = 136 ^ { \circ }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-4_709_821_1603_703} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure} Calculate the total length of the course for this race.

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.

1

1. State the exact value of \(\tan 300 ^ { \circ }\).
2. Express \(300 ^ { \circ }\) in radians, giving your answer in the form \(k \pi\), where \(k\) is a fraction in its lowest terms.

5 A sector of a circle of radius 5 cm has area \(9 \mathrm {~cm} ^ { 2 }\).
Find, in radians, the angle of the sector.
Find also the perimeter of the sector.

4 A sector of a circle of radius 18.0 cm has arc length 43.2 cm .

1. Find in radians the angle of the sector.
2. Find this angle in degrees, giving your answer to the nearest degree.

6 The angle of a sector of a circle is 2 radians and the length of the arc of the sector is 45 cm .
Find

1. the radius of the circle,
2. the area of the sector.

9 A sector of a circle has an angle of 0.8 radians. The arc length is 5 cm . Calculate the radius of the circle and the area of the sector.

8 Fig. 8 shows a sector of a circle with centre O and radius 6 cm and a chord AB which subtends an angle of 1.8 radians at O . \begin{figure}[h] \end{figure}

1. Calculate the area of the sector OAXB .
2. Calculate the area of the triangle OAB and hence find the area of the shaded segment AXB.

8. \includegraphics[max width=\textwidth, alt={}, center]{27703044-8bb3-4809-9454-ae6774fec060-3_501_492_242_607} The diagram shows a circle of radius 12 cm which passes through the points \(P\) and \(Q\). The chord \(P Q\) subtends an angle of \(120 ^ { \circ }\) at the centre of the circle.

1. Find the exact length of the major arc \(P Q\).
2. Show that the perimeter of the shaded minor segment is given by \(k ( 2 \pi + 3 \sqrt { 3 } ) \mathrm { cm }\), where \(k\) is an integer to be found.
3. Find, to 1 decimal place, the area of the shaded minor segment as a percentage of the area of the circle.

7. \includegraphics[max width=\textwidth, alt={}, center]{de1a3480-0d83-43c2-a5a2-2f117b8a50fd-3_376_892_221_427} The diagram shows a design painted on the wall at a karting track. The sign consists of triangle \(A B C\) and two circular sectors of radius 2 metres and 1 metre with centres \(A\) and \(B\) respectively. Given that \(A B = 7 \mathrm {~m} , A C = 3 \mathrm {~m}\) and \(\angle A C B = 2.2\) radians,

1. find the size of \(\angle A B C\) in radians to 3 significant figures,
2. show that \(\angle B A C = 0.588\) radians to 3 significant figures,
3. find the area of triangle \(A B C\),
4. find the area of the wall covered by the design.

2. \includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-1_588_513_813_593} The diagram shows a circle of radius \(r\) and centre \(O\) in which \(A D\) is a diameter.
The points \(B\) and \(C\) lie on the circle such that \(O B\) and \(O C\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively. Show that the area of the shaded region \(O B C\) is \(\frac { 1 } { 6 } r ^ { 2 } ( 3 \sqrt { 3 } - \pi )\).

4

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-4_765_757_203_764} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
   \end{figure} A boat travels from P to Q and then to R . As shown in Fig. 11.1, Q is 10.6 km from P on a bearing of \(045 ^ { \circ }\). R is 9.2 km from P on a bearing of \(113 ^ { \circ }\), so that angle QPR is \(68 ^ { \circ }\). Calculate the distance and bearing of R from Q .
2. Fig. 11.2 shows the cross-section, EBC, of the rudder of a boat. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-4_527_1474_1452_404} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure} BC is an arc of a circle with centre A and radius 80 cm . Angle \(\mathrm { CAB } = \frac { 2 \pi } { 3 }\) radians.
   EC is an arc of a circle with centre D and radius \(r \mathrm {~cm}\). Angle CDE is a right angle.
   1. Calculate the area of sector ABC .
   2. Show that \(r = 40 \sqrt { 3 }\) and calculate the area of triangle CDA.
   3. Hence calculate the area of cross-section of the rudder. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-5_695_1012_271_600} \captionsetup{labelformat=empty} \caption{Fig. 12}
      \end{figure} A water trough is a prism 2.5 m long. Fig. 12 shows the cross-section of the trough, with the depths in metres at 0.1 m intervals across the trough. The trough is full of water.
      1. Use the trapezium rule with 5 strips to calculate an estimate of the area of cross-section of the trough. Hence estimate the volume of water in the trough.
      2. A computer program models the curve of the base of the trough, with axes as shown and units in metres, using the equation \(y = 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15\), for \(0 \leqslant x \leqslant 0.5\). Calculate \(\int _ { 0 } ^ { 0.5 } \left( 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15 \right) \mathrm { d } x\) and state what this represents.
         Hence find the volume of water in the trough as given by this model.