Quadrilateral with diagonal

5. Figure 2 Diagram NOT accurately drawn Figure 2 shows the plan view of a frame for a flat roof.
The shape of the frame consists of triangle \(A B D\) joined to triangle \(B C D\).
Given that

• \(B D = x \mathrm {~m}\)
• \(C D = ( 1 + x ) \mathrm { m }\)
• \(B C = 5 \mathrm {~m}\)
• angle \(B C D = \theta ^ { \circ }\)
  1. show that \(\cos \theta ^ { \circ } = \frac { 13 + x } { 5 + 5 x }\)

Given also that

4 \includegraphics[max width=\textwidth, alt={}, center]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-2_515_713_1567_715} In the diagram, angle \(B D C = 50 ^ { \circ }\) and angle \(B C D = 62 ^ { \circ }\). It is given that \(A B = 10 \mathrm {~cm} , A D = 20 \mathrm {~cm}\) and \(B C = 16 \mathrm {~cm}\).

1. Find the length of \(B D\).
2. Find angle \(B A D\).

4 \includegraphics[max width=\textwidth, alt={}, center]{387a37c4-0997-484c-8e28-954639169ebe-3_309_1084_269_532} In the diagram, \(A B C D\) is a quadrilateral in which \(A D\) is parallel to \(B C\). It is given that \(A B = 9 , B C = 6\), \(C A = 5\) and \(C D = 15\).

1. Show that \(\cos B C A = - \frac { 1 } { 3 }\), and hence find the value of \(\sin B C A\).
2. Find the angle \(A D C\) correct to the nearest \(0.1 ^ { \circ }\).

7. The diagram 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. Show that \(B D = 3 \sqrt { 7 } \mathrm {~cm}\).
2. Find the size of \(\angle B C D\) in degrees to 1 decimal place.
3. Find the area of quadrilateral \(A B C D\).

2 Fig. 10.1 shows Jean's back garden. This is a quadrilateral ABCD with dimensions as shown. \begin{figure}[h] \end{figure}

1. (A) Calculate AC and angle ACB . Hence calculate AD .
   (B) Calculate the area of the garden.
2. The shape of the fence panels used in the garden is shown in Fig. 10.2. EH is the arc of a sector of a circle with centre at the midpoint, M , of side FG , and sector angle 1.1 radians, as shown. \(\mathrm { FG } = 1.8 \mathrm {~m}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cd507fa5-97b2-4edd-ae37-a58aea1de5ed-2_579_981_1512_567} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
   \end{figure} Calculate the area of one of these fence panels.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of triangle \(A B D\) and triangle \(B C D\) Given that

• \(A D C\) is a straight line
• \(B D = ( x + 3 ) \mathrm { cm }\)
• \(B C = x \mathrm {~cm}\)
• angle \(B D C = 30 ^ { \circ }\)
• angle \(B C D = 140 ^ { \circ }\)
  1. show that \(x = 10.5\) correct to 3 significant figures.

Given also that \(A D = ( x - 2 ) \mathrm { cm }\)

find the length of \(A B\), giving your answer to 3 significant figures.

2 Fig. 2 shows a quadrilateral ABCD . The lengths AB and BC are 5 cm and 6 cm respectively. The angles \(\mathrm { ABC } , \mathrm { ACD }\) and DAC are \(60 ^ { \circ } , 60 ^ { \circ }\) and \(75 ^ { \circ }\) respectively. \begin{figure}[h] \end{figure} Calculate the exact value of the length AD.

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 \(A B C D\).

Fig. 10.1 shows Jean's back garden. This is a quadrilateral ABCD with dimensions as shown. \includegraphics{figure_10.1}

1. 1. Calculate AC and angle ACB. Hence calculate AD. [6]
   2. Calculate the area of the garden. [3]
2. The shape of the fence panels used in the garden is shown in Fig. 10.2. EH is the arc of a sector of a circle with centre at the midpoint, M, of side FG, and sector angle 1.1 radians, as shown. FG = 1.8 m. \includegraphics{figure_10.2} Calculate the area of one of these fence panels. [5]