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

8 In this question you must show detailed reasoning. The diagram shows triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-06_737_1383_456_342} The angles \(C A B\) and \(A B C\) are each \(45 ^ { \circ }\), and angle \(A C B = 90 ^ { \circ }\).
The points \(D\) and \(E\) lie on \(A C\) and \(A B\) respectively. \(A E = D E = 1 , D B = 2\). Angle \(B E D = 90 ^ { \circ }\), angle \(E B D = 30 ^ { \circ }\) and angle \(D B C = 15 ^ { \circ }\).

1. Show that \(B C = \frac { \sqrt { 2 } + \sqrt { 6 } } { 2 }\).
2. By considering triangle \(B C D\), show that \(\sin 15 ^ { \circ } = \frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }\).

5 A triangular prism has a cross section \(A B C\) as shown in the diagram below. Angle \(A B C = 25 ^ { \circ }\) Angle \(A C B = 30 ^ { \circ }\) \(B C = 40\) millimetres. The length of the prism is 300 millimetres.
Calculate the volume of the prism, giving your answer to three significant figures.

1. A plot of land \(O A B\) is in the shape of a sector of a circle with centre \(O\).

Given

• \(O A = O B = 5 \mathrm {~km}\)
• angle \(A O B = 1.2\) radians
  1. find the perimeter of the plot of land.
     (2)

\begin{figure}[h] \end{figure} A point \(P\) lies on \(O B\) such that the line \(A P\) divides the plot of land into two regions \(R _ { 1 }\) and \(R _ { 2 }\) as shown in Figure 2. Given that $$\text { area of } R _ { 1 } = 3 \times \text { area of } R _ { 2 }$$

show that the area of \(R _ { 2 } = 3.75 \mathrm {~km} ^ { 2 }\)

Find the length of \(A P\), giving your answer to the nearest 100 m .

2 At 1200 hours an aircraft, \(A\), sets out to intercept a second aircraft, \(B\), which is 200 km away on a bearing of \(300 ^ { \circ }\) and is flying due east at \(600 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Both aircraft are at the same altitude and continue to fly horizontally.

1. (a) Find the bearing on which \(A\) should fly when travelling at \(800 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
   (b) Find the time at which \(A\) intercepts \(B\) in this case.
2. Find the least steady speed at which \(A\) can fly to intercept \(B\).

2 The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).

1 The diagram shows the triangle \(A B C\). \(A B = 10 \mathrm {~cm} , A C = 7 \mathrm {~cm}\) and angle \(B A C = 100 ^ { \circ }\).

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

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 \(A \neq r P\).

2 \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-2_399_933_968_561} The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).

2 \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-2_403_938_964_559} The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).

3 A triangle \(A B C\) has sides \(A B , B C\) and \(C A\) of lengths \(7 \mathrm {~cm} , 6 \mathrm {~cm}\) and 8 cm respectively.

1. Show that \(\cos A B C = \frac { 1 } { 4 }\).
2. Find the area of triangle \(A B C\).

2 \includegraphics[max width=\textwidth, alt={}, center]{48b63de9-f022-4881-a187-f08e3c7d9f1a-2_399_940_952_561} The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).

2 \includegraphics[max width=\textwidth, alt={}, center]{8a0a6e46-99cf-4217-93ad-5ed6e9d7c4ef-2_401_949_959_557} The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).

The triangle \(A B C\) is such that \(A C = 16 \mathrm {~cm} , A B = 25 \mathrm {~cm}\) and \(A \widehat { B C } = 32 ^ { \circ }\). Find two possible values for the area of the triangle \(A B C\). a) Use the binomial theorem to expand \(( a + \sqrt { b } ) ^ { 4 }\).
b) Hence, deduce an expression in terms of \(a\) and \(b\) for \(( a + \sqrt { b } ) ^ { 4 } + ( a - \sqrt { b } ) ^ { 4 }\). a) The vectors \(\mathbf { u }\) and \(\mathbf { v }\) are defined by \(\mathbf { u } = 9 \mathbf { i } - 40 \mathbf { j }\) and \(\mathbf { v } = 3 \mathbf { i } - 4 \mathbf { j }\). Determine the range of values for \(\mu\) such that \(\mu | \mathbf { v } | > | \mathbf { u } |\).
b) The point \(A\) has position vector \(11 \mathbf { i } - 4 \mathbf { j }\) and the point \(B\) has position vector \(21 \mathbf { i } + \mathbf { j }\). Determine the position vector of the point \(C\), which lies between \(A\) and \(B\), such that \(A C : C B\) is \(2 : 3\). Find the values of \(m\) for which the equation \(4 x ^ { 2 } + 8 x - 8 = m ( 4 x - 3 )\) has real roots. [5]

\includegraphics{figure_4} In the diagram, \(ABCD\) is a parallelogram with \(AB = BD = DC = 10\) cm and angle \(ABD = 0.8\) radians. \(APD\) and \(BQC\) are arcs of circles with centres \(B\) and \(D\) respectively.

1. Find the area of the parallelogram \(ABCD\). [2]
2. Find the area of the complete figure \(ABQCDP\). [2]
3. Find the perimeter of the complete figure \(ABQCDP\). [2]

\includegraphics{figure_2} The diagram shows a triangle \(AOB\) in which \(OA\) is 12 cm, \(OB\) is 5 cm and angle \(AOB\) is a right angle. Point \(P\) lies on \(AB\) and \(OP\) is an arc of a circle with centre \(A\). Point \(Q\) lies on \(AB\) and \(OQ\) is an arc of a circle with centre \(B\).

1. Show that angle \(BAO\) is 0.3948 radians, correct to 4 decimal places. [1]
2. Calculate the area of the shaded region. [5]

\includegraphics{figure_6} The diagram shows a triangle \(ABC\) in which \(BC = 20\) cm and angle \(ABC = 90°\). The perpendicular from \(B\) to \(AC\) meets \(AC\) at \(D\) and \(AD = 9\) cm. Angle \(BCA = \theta°\).

1. By expressing the length of \(BD\) in terms of \(\theta\) in each of the triangles \(ABD\) and \(DBC\), show that \(20\sin^2 \theta = 9\cos \theta\). [4]
2. Hence, showing all necessary working, calculate \(\theta\). [3]

\includegraphics{figure_8} The diagram shows an isosceles triangle \(ACB\) in which \(AB = BC = 8\) cm and \(AC = 12\) cm. The arc \(XC\) is part of a circle with centre \(A\) and radius \(12\) cm, and the arc \(YC\) is part of a circle with centre \(B\) and radius \(8\) cm. The points \(A\), \(B\), \(X\) and \(Y\) lie on a straight line.

1. Show that angle \(CBY = 1.445\) radians, correct to \(4\) significant figures. [3]
2. Find the perimeter of the shaded region. [4]

\includegraphics{figure_1} Figure 1 shows the triangle \(ABC\), with \(AB = 8\) cm, \(AC = 11\) cm and \(\angle BAC = 0.7\) radians. The arc \(BD\), where \(D\) lies on \(AC\), 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 \(BC\) and \(CD\) and the arc \(BD\). Find

1. the length of the arc \(BD\), [2]
2. the perimeter of \(R\), giving your answer to 3 significant figures, [4]
3. the area of \(R\), giving your answer to 3 significant figures. [5]

\includegraphics{figure_2} In Figure 2 \(OAB\) is a sector of a circle, radius 5 m. The chord \(AB\) is 6 m long.

1. Show that \(\cos A\hat{O}B = \frac{7}{25}\). [2]
2. Hence find the angle \(A\hat{O}B\) in radians, giving your answer to 3 decimal places. [1]
3. Calculate the area of the sector \(OAB\). [2]
4. Hence calculate the shaded area. [3]

\includegraphics{figure_1} Figure 1 shows 3 yachts \(A\), \(B\) and \(C\) which are assumed to be in the same horizontal plane. Yacht \(B\) is 500 m due north of yacht \(A\) and yacht \(C\) is 700 m from \(A\). The bearing of \(C\) from \(A\) is 015°.

1. Calculate the distance between yacht \(B\) and yacht \(C\), in metres to 3 significant figures. [3]

The bearing of yacht \(C\) from yacht \(B\) is \(\theta°\), as shown in Figure 1.

1. Calculate the value of \(\theta\). [4]

With respect to a fixed origin \(O\), the line \(l_1\) is given by the equation $$\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + \lambda(8\mathbf{i} - \mathbf{j} + 4\mathbf{k})$$ where \(\lambda\) is a scalar parameter. The point \(A\) lies on \(l_1\) Given that \(|\overrightarrow{OA}| = 5\sqrt{10}\)

1. show that at \(A\) the parameter \(\lambda\) satisfies $$81\lambda^2 + 52\lambda - 220 = 0$$ [3]

Hence

1. 1. show that one possible position vector for \(A\) is \(-15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\)
   2. find the other possible position vector for \(A\). [3]

The line \(l_2\) is parallel to \(l_1\) and passes through \(O\). Given that • \(\overrightarrow{OA} = -15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\) • point \(B\) lies on \(l_2\) where \(|\overrightarrow{OB}| = 4\sqrt{10}\)

1. find the area of triangle \(OAB\), giving your answer to one decimal place. [4]

The triangle \(ABC\), shown in the diagram, is such that \(AB = 7\) cm, \(AC = 5\) cm, \(BC = 8\) cm and angle \(ABC = \theta\). \includegraphics{figure_1}

1. Show that \(\theta = 38.2°\), correct to the nearest \(0.1°\). [3]
2. Calculate the area of triangle \(ABC\), giving your answer, in cm\(^2\), to three significant figures. [2]

\includegraphics{figure_2} A sector \(OAB\) of a circle of radius \(r\) cm has angle \(\theta\) radians. The length of the arc of the sector is 12 cm and the area of the sector is 36 cm² (see diagram).

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

\includegraphics{figure_4} In the diagram, \(ABCD\) is a quadrilateral in which \(AD\) is parallel to \(BC\). It is given that \(AB = 9\), \(BC = 6\), \(CA = 5\) and \(CD = 15\).

1. Show that \(\cos BCA = -\frac{1}{3}\), and hence find the value of \(\sin BCA\). [4]
2. Find the angle \(ADC\) correct to the nearest \(0.1°\). [4]

In a triangle \(ABC\), \(AB = 5\sqrt{2}\) cm, \(BC = 8\) cm and angle \(B = 60°\).

1. Find the exact area of the triangle, giving your answer as simply as possible. [3]
2. Find the length of \(AC\), correct to 3 significant figures. [3]