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

The points \(C\) and \(D\) are at a distance \(( 2 \sqrt { } 3 ) a\) apart on a horizontal surface. A rough peg \(A\) is fixed at a vertical distance \(6 a\) above \(C\) and a smooth peg \(B\) is fixed at a vertical distance \(4 a\) above \(D\). A uniform rectangular frame \(P Q R S\), with \(P Q = 3 a\) and \(Q R = 6 a\), is made of rigid thin wire and has weight \(W\). It rests in equilibrium in a vertical plane with \(P S\) on \(A\) and \(S R\) on \(B\), and with angle \(S A C = 30 ^ { \circ }\) (see diagram).

1. Show that \(A B = 4 a\) and that angle \(S A B = 30 ^ { \circ }\).
2. Show that the normal reaction at \(A\) is \(\frac { 1 } { 2 } W\).
3. Find the frictional force at \(A\).

The points \(C\) and \(D\) are at a distance \(( 2 \sqrt { } 3 ) a\) apart on a horizontal surface. A rough peg \(A\) is fixed at a vertical distance \(6 a\) above \(C\) and a smooth peg \(B\) is fixed at a vertical distance \(4 a\) above \(D\). A uniform rectangular frame \(P Q R S\), with \(P Q = 3 a\) and \(Q R = 6 a\), is made of rigid thin wire and has weight \(W\). It rests in equilibrium in a vertical plane with \(P S\) on \(A\) and \(S R\) on \(B\), and with angle \(S A C = 30 ^ { \circ }\) (see diagram).

1. Show that \(A B = 4 a\) and that angle \(S A B = 30 ^ { \circ }\).
2. Show that the normal reaction at \(A\) is \(\frac { 1 } { 2 } W\).
3. Find the frictional force at \(A\).

10

1. \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-7_599_780_267_328} The diagram shows triangle \(A B C\). The perpendicular from \(C\) to \(A B\) meets \(A B\) at \(D\). Angle \(A C D = x\), angle \(D C B = y\), length \(B C = a\) and length \(A C = b\).
   1. Explain why the length of \(C D\) can be written as \(a \cos y\).
   2. Show that the area of the triangle \(A D C\) is given by \(\frac { 1 } { 2 } a b \sin x \cos y\).
   3. Hence, or otherwise, show that \(\sin ( x + y ) = \sin x \cos y + \cos x \sin y\).
2. Given that \(\sin \left( 30 ^ { \circ } + \alpha \right) = \cos \left( 45 ^ { \circ } - \alpha \right)\), show that \(\tan \alpha = 2 + \sqrt { 6 } - \sqrt { 3 } - \sqrt { 2 }\).

1 In the triangle \(A B C\), the length \(A B = 6 \mathrm {~cm}\), the length \(A C = 15 \mathrm {~cm}\) and the angle \(B A C = 30 ^ { \circ }\).

1. Calculate the length \(B C\). \(D\) is the point on \(A C\) such that the length \(B D = 4 \mathrm {~cm}\).
2. Calculate the possible values of the angle \(A D B\).

6. Figure 1 Figure 1 shows a sketch of a triangle \(A B C\) with \(A B = 3 x \mathrm {~cm} , A C = 2 x \mathrm {~cm}\) and angle \(C A B = 60 ^ { \circ }\) Given that the area of triangle \(A B C\) is \(18 \sqrt { 3 } \mathrm {~cm} ^ { 2 }\)

1. show that \(x = 2 \sqrt { 3 }\)
2. Hence find the exact length of BC, giving your answer as a simplified surd.

5. \begin{figure}[h] \end{figure} Figure 1 shows the design for a structure used to support a roof.
The structure consists of four steel beams, \(A B , B D , B C\) and \(A D\).
Given \(A B = 12 \mathrm {~m} , B C = B D = 7 \mathrm {~m}\) and angle \(B A C = 27 ^ { \circ }\)

1. find, to one decimal place, the size of angle \(A C B\). The steel beams can only be bought in whole metre lengths.
2. Find the minimum length of steel that needs to be bought to make the complete structure.

1. A parallelogram \(P Q R S\) has area \(50 \mathrm {~cm} ^ { 2 }\)

Given

• \(P Q\) has length 14 cm
• \(Q R\) has length 7 cm
• angle \(S P Q\) is obtuse
  find
  1. the size of angle \(S P Q\), in degrees, to 2 decimal places,
  2. the length of the diagonal \(S Q\), in cm , to one decimal place.

6 \includegraphics[max width=\textwidth, alt={}, center]{68f1107f-f188-4698-934e-8fd593b25418-4_442_661_840_260} The diagram shows triangle \(A B C\), with \(A B = x \mathrm {~cm} , A C = y \mathrm {~cm}\) and angle \(B A C = 60 ^ { \circ }\). It is given that the area of the triangle is \(( x + y ) \sqrt { 3 } \mathrm {~cm} ^ { 2 }\).

1. Show that \(4 x + 4 y = x y\). When the vertices of the triangle are placed on the circumference of a circle, \(A C\) is a diameter of the circle.
2. Determine the value of \(x\) and the value of \(y\).

1 In the triangle \(A B C , A B = 3 , B C = 4\) and angle \(A B C = 30 ^ { \circ }\). Find the following.

1. The area of the triangle.
2. The length \(A C\).
3. The angle \(A C B\).

1 In triangle \(A B C , A B = 20 \mathrm {~cm}\) and angle \(B = 45 ^ { \circ }\).

1. Given that \(A C = 16 \mathrm {~cm}\), find the two possible values for angle \(C\), correct to 1 decimal place.
2. Given instead that the area of the triangle is \(75 \sqrt { 2 } \mathrm {~cm} ^ { 2 }\), find \(B C\).

4 A triangle ABC has sides \(\mathrm { AB } = 5 \mathrm {~cm} , \mathrm { AC } = 9 \mathrm {~cm}\) and \(\mathrm { BC } = 10 \mathrm {~cm}\).

1. Find the cosine of angle BAC, giving your answer as a fraction in its lowest terms.
2. Find the exact area of the triangle.

5 The diagram shows the triangle ABC in which \(\mathrm { AC } = 13 \mathrm {~cm}\) and AB is the shortest side. The perimeter of the triangle is 32 cm . The area is \(24 \mathrm {~cm} ^ { 2 }\) and \(\sin \mathrm { B } = \frac { 4 } { 5 }\). Determine the lengths of AB and BC .

2 Fig. 2 shows a triangle with one angle of \(117 ^ { \circ }\) given. The lengths are given in centimetres. \begin{figure}[h] \end{figure} Calculate the area of the triangle, giving your answer correct to 3 significant figures.

10 In this question you must show detailed reasoning.
The diagram shows triangle ABC , where \(\mathrm { AB } = 3.9 \mathrm {~cm} , \mathrm { BC } = 4.5 \mathrm {~cm}\) and \(\mathrm { AC } = 3.5 \mathrm {~cm}\). Determine the area of triangle ABC .

5 A triangular field has sides of length \(100 \mathrm {~m} , 120 \mathrm {~m}\) and 135 m .

1. Find the area of the field.
2. Explain why it would not be reasonable to expect your answer in (a) to be accurate to the nearest square metre.

1 Fig. 1 shows triangle \(A B C\). Fig. 1 Calculate the area of triangle \(A B C\), giving your answer correct to 3 significant figures.

8 A circle with centre \(A\) and radius 8 cm and a circle with centre \(C\) and radius 12 cm intersect at points B and D . Quadrilateral \(A B C D\) has area \(60 \mathrm {~cm} ^ { 2 }\).
Determine the two possible values for the length AC.

4 The triangle \(A B C\), shown in the diagram, is such that \(A C = 8 \mathrm {~cm} , C B = 12 \mathrm {~cm}\) and angle \(A C B = \theta\) radians. The area of triangle \(A B C = 20 \mathrm {~cm} ^ { 2 }\).

1. Show that \(\theta = 0.430\) correct to three significant figures.
2. Use the cosine rule to calculate the length of \(A B\), giving your answer to two significant figures.
3. The point \(D\) lies on \(C B\) such that \(A D\) is an arc of a circle centre \(C\) and radius 8 cm . The region bounded by the arc \(A D\) and the straight lines \(D B\) and \(A B\) is shaded in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{48c5470e-6489-4b25-98a6-1b4e101ab01c-004_417_883_1436_557} Calculate, to two significant figures:
   1. the length of the \(\operatorname { arc } A D\);
   2. the area of the shaded region.

4 The triangle \(A B C\), shown in the diagram, is such that \(A C = 8 \mathrm {~cm} , C B = 12 \mathrm {~cm}\) and angle \(A C B = \theta\) radians. The area of triangle \(A B C = 20 \mathrm {~cm} ^ { 2 }\).

1. Show that \(\theta = 0.430\) correct to three significant figures.
2. Use the cosine rule to calculate the length of \(A B\), giving your answer to two significant figures.
3. The point \(D\) lies on \(C B\) such that \(A D\) is an arc of a circle centre \(C\) and radius 8 cm . The region bounded by the arc \(A D\) and the straight lines \(D B\) and \(A B\) is shaded in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9fee4b6f-06e2-4ed8-8835-33ef33b98c94-3_424_894_1434_555} Calculate, to two significant figures:
   1. the length of the \(\operatorname { arc } A D\);
   2. the area of the shaded region.

3 The diagram shows a triangle \(A B C\). The length of \(A C\) is 18.7 cm , and the sizes of angles \(B A C\) and \(A B C\) are \(72 ^ { \circ }\) and \(50 ^ { \circ }\) respectively.

1. Show that the length of \(B C = 23.2 \mathrm {~cm}\), correct to the nearest 0.1 cm .
2. Calculate the area of triangle \(A B C\), giving your answer to the nearest \(\mathrm { cm } ^ { 2 }\).

3 The diagram shows a triangle \(A B C\). The size of angle \(A\) is \(63 ^ { \circ }\), and the lengths of \(A B\) and \(A C\) are 7.4 m and 5.26 m respectively.

1. Calculate the area of triangle \(A B C\), giving your answer in \(\mathrm { m } ^ { 2 }\) to three significant figures.
2. Show that the length of \(B C\) is 6.86 m , correct to three significant figures.
3. Find the value of \(\sin \boldsymbol { B }\) to two significant figures.

3 The triangle \(A B C\), shown in the diagram, is such that \(A B = 5 \mathrm {~cm} , A C = 8 \mathrm {~cm}\), \(B C = 10 \mathrm {~cm}\) and angle \(B A C = \theta\).

1. Show that \(\theta = 97.9 ^ { \circ }\), correct to the nearest \(0.1 ^ { \circ }\).
2. 1. Calculate the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
   2. The line through \(A\), perpendicular to \(B C\), meets \(B C\) at the point \(D\). Calculate the length of \(A D\), giving your answer, in cm , to three significant figures.

4 The triangle \(A B C\), shown in the diagram, is such that \(A B\) is 10 metres and angle \(B A C\) is \(150 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-3_323_746_406_648} The area of triangle \(A B C\) is \(40 \mathrm {~m} ^ { 2 }\).

1. Show that the length of \(A C\) is 16 metres.
2. Calculate the length of \(B C\), giving your answer, in metres, to two decimal places.
3. Calculate the smallest angle of triangle \(A B C\), giving your answer to the nearest \(0.1 ^ { \circ }\).

3 The diagram shows a triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{bfe96138-9587-4efb-95c5-84c4d5eadfbe-3_273_622_356_708} The lengths of \(A C\) and \(B C\) are 5 cm and 6 cm respectively.
The area of triangle \(A B C\) is \(12.5 \mathrm {~cm} ^ { 2 }\), and angle \(A C B\) is obtuse.

1. Find the size of angle \(A C B\), giving your answer to the nearest \(0.1 ^ { \circ }\).
2. Find the length of \(A B\), giving your answer to two significant figures.