Sector with attached triangle

5. \begin{figure}[h] \end{figure} Figure 3 shows the plan view of a viewing platform at a tourist site. The shape of the viewing platform consists of a sector \(A B C O A\) of a circle, centre \(O\), joined to a triangle \(A O D\). Given that

• \(O A = O C = 6 \mathrm {~m}\)
• \(A D = 14 \mathrm {~m}\)
• angle \(A D C = 0.43\) radians
• angle \(A O D\) is an obtuse angle
• \(O C D\) is a straight line
  find
  1. the size of angle \(A O D\), in radians, to 3 decimal places,
  2. the length of arc \(A B C\), in metres, to one decimal place,
  3. the total area of the viewing platform, in \(\mathrm { m } ^ { 2 }\), to one decimal place.

6. \begin{figure}[h] \end{figure} Diagram NOT accurately drawn Figure 1 shows the plan view for the design of a stage.
The design consists of a sector \(O B C\) of a circle, with centre \(O\), joined to two congruent triangles \(O A B\) and \(O D C\). Given that

• angle \(B O C = 2.4\) radians
• area of sector \(B O C = 40 \mathrm {~m} ^ { 2 }\)
• \(A O D\) is a straight line of length 12.5 m
  1. find the radius of the sector, giving your answer, in m , to 2 decimal places,
  2. find the size of angle \(A O B\), in radians, to 2 decimal places.

Hence find

the total area of the stage, giving your answer, in \(\mathrm { m } ^ { 2 }\), to one decimal place,

the total perimeter of the stage, giving your answer, in m , to one decimal place.

7. \begin{figure}[h] \end{figure} The shape \(A B C D A\) consists of a sector \(A B C O A\) of a circle, centre \(O\), joined to a triangle \(A O D\), as shown in Figure 2. The point \(D\) lies on \(O C\).
The radius of the circle is 6 cm , length \(A D\) is 5 cm and angle \(A O D\) is 0.7 radians.

1. Find the area of the sector \(A B C O A\), giving your answer to one decimal place. Given angle \(A D O\) is obtuse,
2. find the size of angle \(A D O\), giving your answer to 3 decimal places.
3. Hence find the perimeter of shape \(A B C D A\), giving your answer to one decimal place.
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3. \begin{figure}[h] \end{figure} Figure 1 shows the design for a badge.
The design consists of two congruent triangles, \(A O C\) and \(B O C\), joined to a sector \(A O B\) of a circle centre \(O\).

• Angle \(A O B = \alpha\)
• \(A O = O B = 3 \mathrm {~cm}\)
• \(O C = 5 \mathrm {~cm}\)

Given that the area of sector \(A O B\) is \(7.2 \mathrm {~cm} ^ { 2 }\)

1. show that \(\alpha = 1.6\) radians.
2. Hence find
   1. the area of the badge, giving your answer in \(\mathrm { cm } ^ { 2 }\) to 2 significant figures,
   2. the perimeter of the badge, giving your answer in cm to one decimal place.

      VIXV SIHIANI III IM IONOO

      VIAV SIHI NI JYHAM ION OO

      VI4V SIHI NI JLIYM ION OO

8. \section*{Diagram NOT to scale} \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a design for a pond.
The design consists of a sector \(A O B X\) of a circle centre \(O\) joined to a quadrilateral \(A O B C\).

• \(B C = 8 \mathrm {~m}\)
• \(O A = O B = 3 \mathrm {~m}\)
• angle \(A O B\) is \(\frac { 2 \pi } { 3 }\) radians
• angle \(B C A\) is \(\frac { \pi } { 6 }\) radians
  1. Calculate (i) the exact area of the sector \(A O B X\),
     (ii) the exact perimeter of the sector \(A O B X\).
  2. Calculate the exact area of the triangle \(A O B\).
  3. Show that the length \(A B\) is \(3 \sqrt { 3 } \mathrm {~m}\).
  4. Find the total surface area of the pond. Give your answer in \(\mathrm { m } ^ { 2 }\) correct to 2 significant figures.

10. \begin{figure}[h] \end{figure} Figure 1 shows the design for a shop sign \(A B C D A\). The sign consists of a triangle \(A O D\) joined to a sector of a circle \(D O B C D\) with radius 1.8 m and centre \(O\). The points \(A , B\) and \(O\) lie on a straight line.
Given that \(A D = 3.9 \mathrm {~m}\) and angle \(B O D\) is 0.84 radians,

1. calculate the size of angle \(D A O\), giving your answer in radians to 3 decimal places.
2. Show that, to one decimal place, the length of \(A O\) is 4.9 m .
3. Find, in \(\mathrm { m } ^ { 2 }\), the area of the shop sign, giving your answer to one decimal place.
4. Find, in m , the perimeter of the shop sign, giving your answer to one decimal place.

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.

7. \begin{figure}[h] \end{figure} Figure 2 shows \(A B C\), a sector of a circle of radius 6 cm with centre \(A\). Given that the size of angle \(B A C\) is 0.95 radians, find

1. the length of the \(\operatorname { arc } B C\),
2. the area of the sector \(A B C\). The point \(D\) lies on the line \(A C\) and is such that \(A D = B D\). The region \(R\), shown shaded in Figure 2, is bounded by the lines \(C D , D B\) and the \(\operatorname { arc } B C\).
3. Show that the length of \(A D\) is 5.16 cm to 3 significant figures. Find
4. the perimeter of \(R\),
5. the area of \(R\), giving your answer to 2 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.

8. \begin{figure}[h] \end{figure} Figure 1 shows the plan view of a stage.
The plan view shows two congruent triangles \(A B O\) and \(G F O\) joined to a sector \(O C D E O\) of a circle, centre \(O\), where

• angle \(C O E = 2.3\) radians
• arc length \(C D E = 27.6 \mathrm {~m}\)
• \(A O G\) is a straight line of length 15 m
  1. Show that \(O C = 12 \mathrm {~m}\).
  2. Show that the size of angle \(A O B\) is 0.421 radians correct to 3 decimal places.

Given that the total length of the front of the stage, \(B C D E F\), is 35 m ,

find the total area of the stage, giving your answer to the nearest square metre.

\includegraphics{figure_8} The diagram shows a symmetrical plate \(ABCDEF\). The line \(ABCD\) is straight and the length of \(BC\) is 2cm. Each of the two sectors \(ABF\) and \(DCE\) is of radius \(r\)cm and each of the angles \(ABF\) and \(DCE\) is equal to \(\frac{1}{4}\pi\) radians.

1. It is given that \(r = 0.4\)cm.
   1. Show that the length \(EF = 2.4\)cm. [2]
   2. Find the area of the plate. Give your answer correct to 3 significant figures. [4]
2. It is given instead that the perimeter of the plate is 6cm. Find the value of \(r\). Give your answer correct to 3 significant figures. [4]

\includegraphics{figure_1} Figure 1 shows the plan view of a design for a stage at a concert. The stage is modelled as a sector \(BCDF\), of a circle centre \(F\), joined to two congruent triangles \(ABF\) and \(EDF\). Given that - \(AFE\) is a straight line - \(AF = FE = 10.7\)m - \(BF = FD = 9.2\)m - angle \(BFD = 1.82\) radians find

1. the perimeter of the stage, in metres, to one decimal place, [5]
2. the area of the stage, in m², to one decimal place. [4]

\includegraphics{figure_1} Figure 1 shows the plan view of a design for a stage at a concert. The stage is modelled as a sector \(BCDF\), of a circle centre \(F\), joined to two congruent triangles \(ABF\) and \(EDF\). Given that \(AFE\) is a straight line, \(AF = FE = 10.7\) m, \(BF = FD = 9.2\) m and angle \(BFD = 1.82\) radians, find the perimeter of the stage, in metres, to one decimal place. [4]