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

11. \begin{figure}[h] \end{figure} Figure 1 shows a triangle \(X Y Z\) with \(X Y = 10 \mathrm {~cm} , Y Z = 16 \mathrm {~cm}\) and \(Z X = 12 \mathrm {~cm}\).

1. Find the size of the angle \(Y X Z\), giving your answer in radians to 3 significant figures. The point \(A\) lies on the line \(X Y\) and the point \(B\) lies on the line \(X Z\) and \(A X = B X = 5 \mathrm {~cm} . A B\) is the arc of a circle with centre \(X\). The shaded region \(S\), shown in Figure 1, is bounded by the lines \(B Z , Z Y , Y A\) and the arc \(A B\). Find
2. the perimeter of the shaded region to 3 significant figures,
3. the area of the shaded region to 3 significant figures.

15. \begin{figure}[h] \end{figure} Figure 5 shows the design for a logo.
The logo is in the shape of an equilateral triangle \(A B C\) of side length \(2 r \mathrm {~cm}\), where \(r\) is a constant. The points \(L , M\) and \(N\) are the midpoints of sides \(A C , A B\) and \(B C\) respectively.
The shaded section \(R\), of the logo, is bounded by three curves \(M N , N L\) and \(L M\). The curve \(M N\) is the arc of a circle centre \(L\), radius \(r \mathrm {~cm}\).
The curve \(N L\) is the arc of a circle centre \(M\), radius \(r \mathrm {~cm}\).
The curve \(L M\) is the arc of a circle centre \(N\), radius \(r \mathrm {~cm}\). Find, in \(\mathrm { cm } ^ { 2 }\), the area of \(R\). Give your answer in the form \(k r ^ { 2 }\), where \(k\) is an exact constant to be determined.

13. \begin{figure}[h] \end{figure} Figure 4 shows the position of two stationary boats, \(A\) and \(B\), and a port \(P\) which are assumed to be in the same horizontal plane. Boat \(A\) is 8.7 km on a bearing of \(314 ^ { \circ }\) from port \(P\).
Boat \(B\) is 3.5 km on a bearing of \(052 ^ { \circ }\) from port \(P\).

1. Show that angle \(A P B\) is \(98 ^ { \circ }\)
2. Find the distance of boat \(B\) from boat \(A\), giving your answer to one decimal place.
3. Find the bearing of boat \(B\) from boat \(A\), giving your answer to the nearest degree.

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of a design for a triangular garden \(A B C\). The garden has sides \(B A\) with length \(10 \mathrm {~m} , B C\) with length 6 m and \(C A\) with length 12 m . The point \(D\) lies on \(A C\) such that \(B D\) is an arc of the circle centre \(A\), radius 10 m . A flowerbed \(B C D\) is shown shaded in Figure 2.

1. Find the size of angle \(B A C\), in radians, to 4 decimal places.
2. Find the perimeter of the flowerbed \(B C D\), in m , to 2 decimal places.
3. Find the area of the flowerbed \(B C D\), in \(\mathrm { m } ^ { 2 }\), to 2 decimal places.

8. \begin{figure}[h] \end{figure} The compound shape \(A B C D A\), shown in Figure 1, consists of a triangle \(A B D\) joined along its edge \(B D\) to a sector \(D B C\) of a circle with centre \(B\) and radius 6 cm . The points \(A , B\) and \(C\) lie on a straight line with \(A B = 5 \mathrm {~cm}\) and \(B C = 6 \mathrm {~cm}\). Angle \(D A B = 1.1\) radians.

1. Show that angle \(A B D = 1.20\) radians to 3 significant figures.
2. Find the area of the compound shape, giving your answer to 3 significant figures.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a triangle \(A B C\) with \(A B = 3 x \mathrm {~cm} , A C = x \mathrm {~cm}\) and angle \(C A B = 60 ^ { \circ }\) Given that the area of triangle \(A B C = 24 \sqrt { 3 }\)

1. show that \(x = 4 \sqrt { 2 }\)
2. Hence find the exact length of \(B C\), giving your answer as a simplified surd.

7. \begin{figure}[h] \end{figure} Figure 1 shows the triangle \(A B C\), with \(A B = 8 \mathrm {~cm} , A C = 11 \mathrm {~cm}\) and \(\angle B A C = 0.7\) radians. The \(\operatorname { arc } B D\), where \(D\) lies on \(A C\), is an arc of a circle with centre \(A\) and radius 8 cm . The region \(R\), shown shaded in Figure 1, is bounded by the straight lines \(B C\) and \(C D\) and the \(\operatorname { arc } B D\). Find

1. the length of the \(\operatorname { arc } B D\),
2. the perimeter of \(R\), giving your answer to 3 significant figures,
3. the area of \(R\), giving your answer to 3 significant figures.

9. \begin{figure}[h] \end{figure} Figure 2 shows a plan of a patio. The patio \(P Q R S\) is in the shape of a sector of a circle with centre \(Q\) and radius 6 m . Given that the length of the straight line \(P R\) is \(6 \sqrt { } 3 \mathrm {~m}\),

1. find the exact size of angle \(P Q R\) in radians.
2. Show that the area of the patio \(P Q R S\) is \(12 \pi \mathrm {~m} ^ { 2 }\).
3. Find the exact area of the triangle \(P Q R\).
4. Find, in \(\mathrm { m } ^ { 2 }\) to 1 decimal place, the area of the segment \(P R S\).
5. Find, in \(m\) to 1 decimal place, the perimeter of the patio \(P Q R S\).

7. \begin{figure}[h] \end{figure} The shape \(B C D\) shown in Figure 3 is a design for a logo. The straight lines \(D B\) and \(D C\) are equal in length. The curve \(B C\) is an arc of a circle with centre \(A\) and radius 6 cm . The size of \(\angle B A C\) is 2.2 radians and \(A D = 4 \mathrm {~cm}\). Find

1. the area of the sector \(B A C\), in \(\mathrm { cm } ^ { 2 }\),
2. the size of \(\angle D A C\), in radians to 3 significant figures,
3. the complete area of the logo design, to the nearest \(\mathrm { cm } ^ { 2 }\).

4. \begin{figure}[h] \end{figure} An emblem, as shown in Figure 1, consists of a triangle \(A B C\) joined to a sector \(C B D\) of a circle with radius 4 cm and centre \(B\). The points \(A , B\) and \(D\) lie on a straight line with \(A B = 5 \mathrm {~cm}\) and \(B D = 4 \mathrm {~cm}\). Angle \(B A C = 0.6\) radians and \(A C\) is the longest side of the triangle \(A B C\).

1. Show that angle \(A B C = 1.76\) radians, correct to 3 significant figures.
2. Find the area of the emblem.

2. In the triangle \(A B C , A B = 11 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and \(C A = 8 \mathrm {~cm}\).

1. Find the size of angle \(C\), giving your answer in radians to 3 significant figures.
2. Find the area of triangle \(A B C\), giving your answer in \(\mathrm { cm } ^ { 2 }\) to 3 significant figures.

8. \begin{figure}[h] \end{figure} Figure 2 shows the design for a triangular garden \(A B C\) where \(A B = 7 \mathrm {~m} , A C = 13 \mathrm {~m}\) and \(B C = 10 \mathrm {~m}\). Given that angle \(B A C = \theta\) radians,

1. show that, to 3 decimal places, \(\theta = 0.865\) The point \(D\) lies on \(A C\) such that \(B D\) is an arc of the circle centre \(A\), radius 7 m .
   The shaded region \(S\) is bounded by the arc \(B D\) and the lines \(B C\) and \(D C\). The shaded region \(S\) will be sown with grass seed, to make a lawned area. Given that 50 g of grass seed are needed for each square metre of lawn,
2. find the amount of grass seed needed, giving your answer to the nearest 10 g .

5. \begin{figure}[h] \end{figure} The shape \(A B C D E A\), as shown in Figure 2, consists of a right-angled triangle \(E A B\) and a triangle \(D B C\) joined to a sector \(B D E\) of a circle with radius 5 cm and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(B C = 7.5 \mathrm {~cm}\).
Angle \(E A B = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.4\) radians and \(C D = 6.1 \mathrm {~cm}\).

1. Find, in \(\mathrm { cm } ^ { 2 }\), the area of the sector \(B D E\).
2. Find the size of the angle \(D B C\), giving your answer in radians to 3 decimal places.
3. Find, in \(\mathrm { cm } ^ { 2 }\), the area of the shape \(A B C D E A\), giving your answer to 3 significant figures.

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.

9. Figure 3 $$( x + 1 ) ^ { 2 }$$ Figure 3 shows a triangle \(P Q R\). The size of angle \(Q P R\) is \(30 ^ { \circ }\), the length of \(P Q\) is \(( x + 1 )\) and the length of \(P R\) is \(( 4 - x ) ^ { 2 }\), where \(X \in \Re\).

1. Show that the area \(A\) of the triangle is given by \(A = \frac { 1 } { 4 } \left( x ^ { 3 } - 7 x ^ { 2 } + 8 x + 16 \right)\)
2. Use calculus to prove that the area of \(\triangle P Q R\) is a maximum when \(x = \frac { 2 } { 3 }\). Explain clearly how you know that this value of \(x\) gives the maximum area.
3. Find the maximum area of \(\triangle P Q R\).
4. Find the length of \(Q R\) when the area of \(\triangle P Q R\) is a maximum. END

1. Relative to a fixed origin \(O\),
   the point \(A\) has position vector \(( 2 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k } )\) the point \(B\) has position vector \(( 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } )\) the point \(C\) has position vector ( \(2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\) )

The line \(l\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l\).
3. Show that the size of the angle \(C A B\) is \(62.8 ^ { \circ }\), to one decimal place.
4. Hence find the area of triangle \(C A B\), giving your answer to 3 significant figures. The point \(D\) lies on the line \(l\). Given that the area of triangle \(C A D\) is twice the area of triangle \(C A B\),
5. find the two possible position vectors of point \(D\).
   

4. \begin{figure}[h] \end{figure} Figure 3 shows two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) acting on a particle.
The angle between the direction of \(\mathbf { P }\) and the direction of \(\mathbf { Q }\) is \(150 ^ { \circ }\) Force \(\mathbf { P }\) has magnitude \(X\) newtons.
Force \(\mathbf { Q }\) has magnitude \(5 \sqrt { 3 } \mathrm {~N}\).
The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) has magnitude \(\sqrt { 129 } \mathrm {~N}\).
Find

1. the value of \(X\).
2. the angle between \(\mathbf { Q }\) and the resultant, giving your answer to the nearest degree.

2 Triangle \(A B C\) has \(A B = 10 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and angle \(B = 80 ^ { \circ }\). Calculate

1. the area of the triangle,
2. the length of \(C A\),
3. the size of angle \(C\).

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 \(B C D\).

4 Fig. 4 For triangle ABC shown in Fig. 4, calculate

1. the length of BC ,
2. the area of triangle ABC .

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.

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.

10 At 1200 the captain of a ship observes that the bearing of a lighthouse is \(340 ^ { \circ }\). His position is at A.
At 1230 he takes another bearing of the lighthouse and finds it to be \(030 ^ { \circ }\). During this time the ship moves on a constant course of \(280 ^ { \circ }\) to the point B . His plot on the chart is as shown in Fig. 11 below. \begin{figure}[h] \end{figure}

1. Write down the size of the angles LAB and LBA .
2. The captain believes that at A he is 5 km from L . Assuming that LA is exactly 5 km , show that LB is 4.61 km , correct to 2 decimal places, and find AB . Hence calculate the speed of the ship.
3. The speed of the ship is actually 10 kilometres per hour. Given that the bearings of \(340 ^ { \circ }\) and \(030 ^ { \circ }\) and the ship's course of \(280 ^ { \circ }\) are all accurate, calculate the true value of the distance LA.