Multiple circles or sectors

5 \includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-06_720_686_260_733} In the diagram, \(X\) and \(Y\) are points on the line \(A B\) such that \(B X = 9 \mathrm {~cm}\) and \(A Y = 11 \mathrm {~cm}\). Arc \(B C\) is part of a circle with centre \(X\) and radius 9 cm , where \(C X\) is perpendicular to \(A B\). Arc \(A C\) is part of a circle with centre \(Y\) and radius 11 cm .

1. Show that angle \(X Y C = 0.9582\) radians, correct to 4 significant figures.
2. Find the perimeter of \(A B C\).

6 \includegraphics[max width=\textwidth, alt={}, center]{3fd0b68f-41b1-4eee-8018-bcaf3cf22950-3_801_1273_255_434} The diagram shows a circle \(C _ { 1 }\) touching a circle \(C _ { 2 }\) at a point \(X\). Circle \(C _ { 1 }\) has centre \(A\) and radius 6 cm , and circle \(C _ { 2 }\) has centre \(B\) and radius 10 cm . Points \(D\) and \(E\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively and \(D E\) is parallel to \(A B\). Angle \(D A X = \frac { 1 } { 3 } \pi\) radians and angle \(E B X = \theta\) radians.

1. By considering the perpendicular distances of \(D\) and \(E\) from \(A B\), show that the exact value of \(\theta\) is \(\sin ^ { - 1 } \left( \frac { 3 \sqrt { } 3 } { 10 } \right)\).
2. Find the perimeter of the shaded region, correct to 4 significant figures.

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the plan view of a platform.
The plan view of the platform consists of a sector \(D O C\) of a circle centre \(O\) joined to a sector \(A O B E A\) of a different circle, also with centre \(O\). Given that

• angle \(A O B = 0.8\) radians
• arc length \(C D = 9 \mathrm {~m}\)
• \(D A : A O = 3 : 5\)
  1. show that \(A O = 7.03 \mathrm {~m}\) to 3 significant figures.
  2. Find the perimeter of the platform, in m , to 3 significant figures.
  3. Find the total area of the platform, giving your answer in \(\mathrm { m } ^ { 2 }\) to the nearest whole number.

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the outline of the face of a ceiling fan viewed from below.
The fan consists of three identical sections congruent to \(O A B C D O\), shown in Figure 3, where

• \(O A B O\) is a sector of a circle with centre \(O\) and radius 9 cm
• \(O B C D O\) is a sector of a circle with centre \(O\) and radius 84 cm
• angle \(A O D = \frac { 2 \pi } { 3 }\) radians

Given that the length of the arc \(A B\) is 15 cm ,

1. show that the length of the arc \(C D\) is 35.9 cm to one decimal place. The face of the fan is modelled to be a flat surface.
   Find, according to the model,
2. the perimeter of the face of the fan, giving your answer to the nearest cm,
3. the surface area of the face of the fan. Give your answer to 3 significant figures and make your units clear.

7 \includegraphics[max width=\textwidth, alt={}, center]{87012792-fa63-4003-875d-b8e7739037f1-3_412_707_751_680} The diagram shows two circles of radius 7 cm with centres \(A\) and \(B\). The distance \(A B\) is 12 cm and the point \(C\) lies on both circles. The region common to both circles is shaded.

1. Show that angle \(C A B\) is 0.5411 radians, correct to 4 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

11

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-5_469_878_274_671} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
   \end{figure} Fig. 11.1 shows the surface ABCD of a TV presenter's desk. AB and CD are arcs of circles with centre O and sector angle 2.5 radians. \(\mathrm { OC } = 60 \mathrm {~cm}\) and \(\mathrm { OB } = 140 \mathrm {~cm}\).
   (A) Calculate the length of the arc CD.
   (B) Calculate the area of the surface ABCD of the desk.
2. The TV presenter is at point P , shown in Fig. 11.2. A TV camera can move along the track EF , which is of length 3.5 m . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-5_378_877_1334_675} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure} When the camera is at E , the TV presenter is 1.6 m away. When the camera is at F , the TV presenter is 2.8 m away.
   (A) Calculate, in degrees, the size of angle EFP.
   (B) Calculate the shortest possible distance between the camera and the TV presenter.

10. \begin{figure}[h] \end{figure} Circle \(C _ { 1 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 64\) with centre \(O _ { 1 }\).
Circle \(C _ { 2 }\) has equation \(( x - 6 ) ^ { 2 } + y ^ { 2 } = 100\) with centre \(O _ { 2 }\).
The circles meet at points \(A\) and \(B\) as shown in Figure 3.
a. Show that angle \(A O _ { 2 } B = 1.85\) radians to 3 significant figures.
(3)
b. Find the area of the shaded region, giving your answer correct to 1 decimal place.

\includegraphics{figure_7} The diagram shows two circles with centres \(A\) and \(B\) having radii 8 cm and 10 cm respectively. The two circles intersect at \(C\) and \(D\) where \(CAD\) is a straight line and \(AB\) is perpendicular to \(CD\).

1. Find angle \(ABC\) in radians. [1]
2. Find the area of the shaded region. [6]

\includegraphics{figure_3} In the diagram, \(CXD\) is a semicircle of radius \(7\) cm with centre \(A\) and diameter \(CD\). The straight line \(YAX\) is perpendicular to \(CD\), and the arc \(CYD\) is part of a circle with centre \(B\) and radius \(8\) cm. Find the total area of the region enclosed by the two arcs. [6]