1.05c Area of triangle: using 1/2 ab sin(C)

1 The diagram shows a triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-2_423_707_612_657} The lengths of \(A C\) and \(B C\) are 5 cm and 4.8 cm respectively.
The size of the angle \(B C A\) is \(30 ^ { \circ }\).

1. Calculate the area of the triangle \(A B C\).
2. Calculate the length of \(A B\), giving your answer to three significant figures.

2 The diagram shows a triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-2_757_558_1409_726} The lengths of \(A C\) and \(B C\) are 4.8 cm and 12 cm respectively.
The size of the angle \(B A C\) is \(100 ^ { \circ }\).

1. Show that angle \(A B C = 23.2 ^ { \circ }\), correct to the nearest \(0.1 ^ { \circ }\).
2. Calculate the area of triangle \(A B C\), giving your answer in \(\mathrm { cm } ^ { 2 }\) to three significant figures.

4 The diagram shows a triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-3_394_522_1062_751} The size of angle \(B A C\) is \(65 ^ { \circ }\), and the lengths of \(A B\) and \(A C\) are 7.6 m and 8.3 m respectively.

1. Show that the length of \(B C\) is 8.56 m , correct to three significant figures.
2. Calculate the area of triangle \(A B C\), giving your answer in \(\mathrm { m } ^ { 2 }\) to three significant figures.
3. The perpendicular from \(A\) to \(B C\) meets \(B C\) at the point \(D\). Calculate the length of \(A D\), giving your answer to the nearest 0.1 m .

3 The triangle \(A B C\), shown in the diagram, is such that \(A B = 6 \mathrm {~cm} , B C = 15 \mathrm {~cm}\), angle \(B A C = 150 ^ { \circ }\) and angle \(A C B = \theta\). \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-4_376_867_406_584}

1. Show that \(\theta = 11.5 ^ { \circ }\), correct to the nearest \(0.1 ^ { \circ }\).
2. Calculate the area of triangle \(A B C\), giving your answer in \(\mathrm { cm } ^ { 2 }\) to three significant figures.

1 The triangle \(A B C\), shown in the diagram, is such that \(A C = 9 \mathrm {~cm} , B C = 10 \mathrm {~cm}\), angle \(A B C = 54 ^ { \circ }\) and the acute angle \(B A C = \theta\).

1. Show that \(\theta = 64 ^ { \circ }\), correct to the nearest degree.
2. Calculate the area of triangle \(A B C\), giving your answer to the nearest square centimetre.

2 The triangle \(A B C\), shown in the diagram, is such that \(A B = 26 \mathrm {~cm}\) and \(B C = 31.5 \mathrm {~cm}\). The acute angle \(A B C\) is \(\theta\), where \(\sin \theta = \frac { 5 } { 13 }\).

1. Calculate the area of triangle \(A B C\).
2. Find the exact value of \(\cos \theta\).
3. Calculate the length of \(A C\).

1 The diagram shows a triangle \(A B C\). The size of angle \(B A C\) is \(47 ^ { \circ }\) and the lengths of \(A B\) and \(A C\) are 5 cm and 12 cm respectively.

1. Calculate the area of the triangle \(A B C\), giving your answer to the nearest \(\mathrm { cm } ^ { 2 }\).
2. Calculate the length of \(B C\), giving your answer, in cm , to one decimal place.
   [0pt] [3 marks]

6 The diagram shows a triangle \(A B C\). The lengths of \(A B , B C\) and \(A C\) are \(8 \mathrm {~cm} , 5 \mathrm {~cm}\) and 9 cm respectively.
Angle \(B A C\) is \(\theta\) radians.

1. Show that \(\theta = 0.586\), correct to three significant figures.
2. Find the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
3. A circular sector, centre \(A\) and radius \(r \mathrm {~cm}\), is removed from triangle \(A B C\). The remaining shape is shown shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{e183578a-29a8-4112-b941-06c8894ed078-14_467_677_1462_685} Given that the area of the sector removed is equal to the area of the shaded shape, find the perimeter of the shaded shape. Give your answer in cm to three significant figures.
   [0pt] [6 marks]

8. Figure 2 Figure 2 shows the quadrilateral \(A B C D\) in which \(A B = 6 \mathrm {~cm} , B C = 3 \mathrm {~cm} , C D = 8 \mathrm {~cm}\), \(A D = 9 \mathrm {~cm}\) and \(\angle B A D = 60 ^ { \circ }\).

1. Using the cosine rule, show that \(B D = 3 \sqrt { 7 } \mathrm {~cm}\).
2. Find the size of \(\angle B C D\) in degrees.
3. Find the area of quadrilateral \(A B C D\).

6. \begin{figure}[h] \end{figure} Figure 1 shows triangle \(A B C\) in which \(A C = 8 \mathrm {~cm}\) and \(\angle B A C = \angle B C A = 30 ^ { \circ }\).

1. Find the area of triangle \(A B C\) in the form \(k \sqrt { 3 }\). The point \(M\) is the mid-point of \(A C\) and the points \(N\) and \(O\) lie on \(A B\) and \(B C\) such that \(M N\) and \(M O\) are arcs of circles with centres \(A\) and \(C\) respectively.
2. Show that the area of the shaded region \(B N M O\) is \(\frac { 8 } { 3 } ( 2 \sqrt { 3 } - \pi ) \mathrm { cm } ^ { 2 }\).

1. \begin{figure}[h] \end{figure} Figure 1 shows triangle \(A B C\) in which \(A B = 12.6 \mathrm {~cm} , \angle A B C = 107 ^ { \circ }\) and \(\angle A C B = 31 ^ { \circ }\).
Find, to 3 significant figures,

1. the length \(B C\),
2. the area of triangle \(A B C\).

9. \begin{figure}[h] \end{figure} Figure 3 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. use the sine rule to 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. The sides of a triangle have lengths of \(7 \mathrm {~cm} , 8 \mathrm {~cm}\) and 10 cm . Find the area of the triangle correct to 3 significant figures.

5. Figure 2 Figure 2 shows triangle \(P Q R\) in which \(P Q = 7\) and \(P R = 3 \sqrt { 5 }\).
Given that \(\sin ( \angle Q P R ) = \frac { 2 } { 3 }\) and that \(\angle Q P R\) is acute,

1. find the exact value of \(\cos ( \angle Q P R )\) in its simplest form,
2. show that \(Q R = 2 \sqrt { 6 }\),
3. find \(\angle P Q R\) in degrees to 1 decimal place.

8 A cricket ball is hit at ground level on a horizontal surface. It initially moves at \(28 \mathrm {~ms} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal.

1. Find the maximum height of the ball during its flight.
2. The ball is caught when it is at a height of 2 metres above ground level, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-5_332_1070_1601_477} Show that the time that it takes for the ball to travel from the point where it was hit to the point where it was caught is 4.28 seconds, correct to three significant figures.
3. Find the speed of the ball when it is caught.

9 A small ball P is projected with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(( \alpha + \theta )\) from a point O at the bottom of a plane inclined at \(\alpha\) to the horizontal. P subsequently hits the plane at a point R , where OR is a line of greatest slope, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-07_456_862_406_242}

1. By deriving an expression, in terms of \(\theta\), \(\alpha\) and \(g\), for the time of flight of P , show that the distance OR, in metres, is $$\frac { 50 \sin \theta \cos ( \theta + \alpha ) } { g \cos ^ { 2 } \alpha }$$
2. By using the identity \(2 \sin \mathrm {~A} \cos \mathrm {~B} \equiv \sin ( \mathrm {~A} + \mathrm { B } ) - \sin ( \mathrm { B } - \mathrm { A } )\), determine, in terms of \(g\) and \(\sin \alpha\), an expression for the maximum range of P up the plane, as \(\theta\) varies.
3. Given that OR is the maximum range of P up the plane and is equal to 1.8 m , determine the value of \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-08_625_1180_255_239} A rigid wire ABC is fixed in a vertical plane. The section AB of the wire, of length \(b\), is straight and horizontal. The section BC of the wire is smooth and in the form of a circular arc of radius \(a\) and length \(\frac { 1 } { 2 } a \pi\). The centre of the arc is O , which is vertically above B . A bead P of mass \(m\) is threaded on the wire and projected from B with speed \(u\) towards C . The angle BOP when P is between B and C is denoted by \(\theta\), as shown in the diagram.

11 In this question you must show detailed reasoning. The diagram shows triangle ABC , with \(\mathrm { BC } = 8 \mathrm {~cm}\) and angle \(\mathrm { BAC } = 45 ^ { \circ }\).
The point D on AC is such that \(\mathrm { DC } = 5 \mathrm {~cm}\) and \(\mathrm { BD } = 7 \mathrm {~cm}\). \includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-7_684_553_1119_258} Determine the exact length of AB .

7 Two lifeboat stations, \(P\) and \(Q\), are situated on the coastline with \(Q\) being due south of \(P\). A stationary ship is at sea, at a distance of 4.8 km from \(P\) and a distance of 2.2 km from \(Q\). The ship is on a bearing of \(155 ^ { \circ }\) from \(P\).

1. Find any possible bearings of the ship from \(Q\).
2. Find the shortest distance from the ship to the line \(P Q\).
3. Give a reason why the actual distance from the ship to the coastline may be different to your answer to part (ii).

4 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-5_487_789_251_639} The diagram shows the triangle \(A O B\), in which angle \(A O B = 0.8\) radians, \(O A = 7 \mathrm {~cm}\) and \(O B = 10 \mathrm {~cm}\). \(C D\) is the arc of a circle with centre \(O\) and radius \(O C\). The area of the triangle \(A O B\) is twice the area of the sector COD

1. Find the length \(O C\).
2. Find the perimeter of the region \(A B C D\).

8. The line \(l _ { 1 }\) passes through the point \(( 9 , - 4 )\) and has gradient \(\frac { 1 } { 3 }\).

1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(l _ { 2 }\) passes through the origin \(O\) and has gradient - 2 . The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).
2. Calculate the coordinates of \(P\). Given that \(l _ { 1 }\) crosses the \(y\)-axis at the point \(C\),
3. calculate the exact area of \(\triangle O C P\). \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_59_2568_1882} \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_1829_2648_114}

4 The triangle \(A B C\), shown in the diagram, is such that \(B C = 6 \mathrm {~cm} , A C = 5 \mathrm {~cm}\) and \(A B = 4 \mathrm {~cm}\). The angle \(B A C\) is \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-3_442_652_452_678}

1. Use the cosine rule to show that \(\cos \theta = \frac { 1 } { 8 }\).
2. Hence use a trigonometrical identity to show that \(\sin \theta = \frac { 3 \sqrt { 7 } } { 8 }\).
3. Hence find the area of the triangle \(A B C\).

3 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 20 cm . The angle between the radii \(O A\) and \(O B\) is \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-3_453_499_429_804} The length of the \(\operatorname { arc } A B\) is 28 cm .

1. Show that \(\theta = 1.4\).
2. Find the area of the sector \(O A B\).
3. The point \(D\) lies on \(O A\). The region bounded by the line \(B D\), the line \(D A\) and the arc \(A B\) is shaded. \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-3_440_380_1372_806} The length of \(O D\) is 15 cm .
   1. Find the area of the shaded region, giving your answer to three significant figures.
      (3 marks)
   2. Use the cosine rule to calculate the length of \(B D\), giving your answer to three significant figures.
      (3 marks)

7. \begin{figure}[h] \end{figure} Figure 4 shows a shape \(S ( \theta )\) made up of five line segments \(A B , B C , C D , D E\) and \(E A\) . The lengths of the sides are \(A B = B C = 5 \mathrm {~cm} , C D = E A = 3 \mathrm {~cm}\) and \(D E = 7 \mathrm {~cm}\) . Angle \(B A E =\) angle \(B C D = \theta\) radians. The length of each line segment always remains the same but the value of \(\theta\) can be varied so that different symmetrical shapes can be formed,with the added restriction that none of the line segments cross.

1. Sketch \(S ( \pi )\) ,labelling the vertices clearly. The shape \(S ( \phi )\) is a trapezium.
2. Sketch \(S ( \phi )\) and calculate the value of \(\phi\) . The smallest possible value for \(\theta\) is \(\alpha\) ,where \(\alpha > 0\) ,and the largest possible value for \(\theta\) is \(\beta\) , where \(\beta > \pi\) .
3. Show that \(\alpha = \arccos \left( \frac { 29 } { 40 } \right) \cdot \left[ \arccos ( x ) \right.\) is an alternative notation for \(\left. \cos ^ { - 1 } ( x ) \right]\)
4. Find the value of \(\beta\) . The area,in \(\mathrm { cm } ^ { 2 }\) ,of shape \(S ( \theta )\) is \(R ( \theta )\) .
5. Show that for \(\alpha \leqslant \theta < \pi\) $$R ( \theta ) = 15 \sin \theta + \frac { 7 } { 4 } \sqrt { 87 - 120 \cos \theta }$$ Given that this formula for \(R ( \theta )\) holds for \(\alpha \leqslant \theta \leqslant \beta\)
6. show that \(R ( \theta )\) has only one stationary point and that this occurs when \(\theta = \frac { 2 \pi } { 3 }\)
7. find the maximum and minimum values of \(R ( \theta )\). FOR STYLE, CLARITY AND PRESENTATION: 7 MARKS TOTAL FOR PAPER: 100 MARKS
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