Sector and arc length

8
\includegraphics[max width=\textwidth, alt={}, center]{80a20f05-61db-42d9-b4ba-53eea2290b2d-10_780_814_264_662} The diagram shows a symmetrical metal plate. The plate is made by removing two identical pieces from a circular disc with centre \(C\). The boundary of the plate consists of two \(\operatorname { arcs } P S\) and \(Q R\) of the original circle and two semicircles with \(P Q\) and \(R S\) as diameters. The radius of the circle with centre \(C\) is 4 cm , and \(P Q = R S = 4 \mathrm {~cm}\) also.

1. Show that angle \(P C S = \frac { 2 } { 3 } \pi\) radians.
2. Find the exact perimeter of the plate.
3. Show that the area of the plate is \(\left( \frac { 20 } { 3 } \pi + 8 \sqrt { 3 } \right) \mathrm { cm } ^ { 2 }\).

4 The diagram shows a sector \(A B C\) of a circle with centre \(A\) and radius 8 cm . The area of the sector is \(\frac { 16 } { 3 } \pi \mathrm {~cm} ^ { 2 }\). The point \(D\) lies on the \(\operatorname { arc } B C\). Find the perimeter of the segment \(B C D\).

6
\includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-07_389_552_267_799} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). Angle \(A O B = \theta\) radians. It is given that the length of the \(\operatorname { arc } A B\) is 9.6 cm and that the area of the sector \(O A B\) is \(76.8 \mathrm {~cm} ^ { 2 }\).

1. Find the area of the shaded region.
2. Find the perimeter of the shaded region.

3
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-05_483_561_287_753} The diagram shows a sector of a circle with centre \(C\). The radii \(C A\) and \(C B\) each have length \(r \mathrm {~cm}\) and the size of the reflex angle \(A C B\) is \(\theta\) radians. The sector, shaded in the diagram, has a perimeter of 65 cm and an area of \(225 \mathrm {~cm} ^ { 2 }\).

1. Find the values of \(r\) and \(\theta\).
2. Find the area of triangle \(A C B\).

4
\includegraphics[max width=\textwidth, alt={}, center]{fe4c3555-5736-48c4-b61a-9f6b9a1ee46e-2_645_652_1023_744} The diagram shows a square \(A B C D\) of side 10 cm . The mid-point of \(A D\) is \(O\) and \(B X C\) is an arc of a circle with centre \(O\).

1. Show that angle \(B O C\) is 0.9273 radians, correct to 4 decimal places.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

2
\includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-2_501_641_461_753} The diagram shows a circle \(C\) with centre \(O\) and radius 3 cm . The radii \(O P\) and \(O Q\) are extended to \(S\) and \(R\) respectively so that \(O R S\) is a sector of a circle with centre \(O\). Given that \(P S = 6 \mathrm {~cm}\) and that the area of the shaded region is equal to the area of circle \(C\),

1. show that angle \(P O Q = \frac { 1 } { 4 } \pi\) radians,
2. find the perimeter of the shaded region.

3 A sector of a circle of radius \(r \mathrm {~cm}\) has an area of \(A \mathrm {~cm} ^ { 2 }\). Express the perimeter of the sector in terms of \(r\) and \(A\).

6
\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-3_293_502_269_826} The diagram shows the sector \(O P Q\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The angle \(P O Q\) is \(\theta\) radians and the perimeter of the sector is 20 cm .

1. Show that \(\theta = \frac { 20 } { r } - 2\).
2. Hence express the area of the sector in terms of \(r\).
3. In the case where \(r = 8\), find the length of the chord \(P Q\).

2
\includegraphics[max width=\textwidth, alt={}, center]{933cdfe1-27bb-450d-8b9a-b494916242cb-2_625_582_397_778} In the diagram, \(O A B\) and \(O C D\) are radii of a circle, centre \(O\) and radius 16 cm . Angle \(A O C = \alpha\) radians. \(A C\) and \(B D\) are arcs of circles, centre \(O\) and radii 10 cm and 16 cm respectively.

1. In the case where \(\alpha = 0.8\), find the area of the shaded region.
2. Find the value of \(\alpha\) for which the perimeter of the shaded region is 28.9 cm .

3
\includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-2_515_750_669_699} In the diagram \(O C A\) and \(O D B\) are radii of a circle with centre \(O\) and radius \(2 r \mathrm {~cm}\). Angle \(A O B = \alpha\) radians. \(C D\) and \(A B\) are arcs of circles with centre \(O\) and radii \(r \mathrm {~cm}\) and \(2 r \mathrm {~cm}\) respectively. The perimeter of the shaded region \(A B D C\) is \(4.4 r \mathrm {~cm}\).

1. Find the value of \(\alpha\).
2. It is given that the area of the shaded region is \(30 \mathrm {~cm} ^ { 2 }\). Find the value of \(r\).
   \(4 C\) is the mid-point of the line joining \(A ( 14 , - 7 )\) to \(B ( - 6,3 )\). The line through \(C\) perpendicular to \(A B\) crosses the \(y\)-axis at \(D\).

4. \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a house \(A B C D\) and a lawn \(A P C D A\).
\(A B C D\) is a rectangle with \(A B = 16 \mathrm {~m}\).
\(A P C O A\) is a sector of a circle centre \(O\) with radius 12 m . The point \(O\) lies on the line \(D C\), as shown in Figure 2.

1. Show that the size of angle \(A O D\) is 1.231 radians to 3 decimal places. The lawn \(A P C D A\) is shown shaded in Figure 2.
2. Find the area of the lawn, in \(\mathrm { m } ^ { 2 }\), to one decimal place.
3. Find the perimeter of the lawn, in metres, to one decimal place.

7. \begin{figure}[h] \end{figure} The line \(l _ { 1 }\) has equation \(4 y + 3 x = 48\)
The line \(l _ { 1 }\) cuts the \(y\)-axis at the point \(C\), as shown in Figure 3.

1. State the \(y\) coordinate of \(C\). The point \(D ( 8,6 )\) lies on \(l _ { 1 }\)
   The line \(l _ { 2 }\) passes through \(D\) and is perpendicular to \(l _ { 1 }\) The line \(l _ { 2 }\) cuts the \(y\)-axis at the point \(E\) as shown in Figure 3.
2. Show that the \(y\) coordinate of \(E\) is \(- \frac { 14 } { 3 }\) A sector \(B C E\) of a circle with centre \(C\) is also shown in Figure 3. Given that angle \(B C E\) is 1.8 radians,
3. find the length of arc \(B E\). The region \(C B E D\), shown shaded in Figure 3, consists of the sector \(B C E\) joined to the triangle \(C D E\).
4. Calculate the exact area of the region \(C B E D\).

3. \begin{figure}[h] \end{figure} The circle \(C\) with centre \(T\) and radius \(r\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 16 y + 139 = 0$$

1. Find the coordinates of the centre of \(C\).
2. Show that \(r = 5\) The line \(L\) has equation \(x = 13\) and crosses \(C\) at the points \(P\) and \(Q\) as shown in Figure 1.
3. Find the \(y\) coordinate of \(P\) and the \(y\) coordinate of \(Q\). Given that, to 3 decimal places, the angle \(P T Q\) is 1.855 radians,
4. find the perimeter of the sector \(P T Q\).

4 A circle has diameter \(d\), circumference \(C\), and area \(A\). Starting with the standard formulae for a circle, show that \(C d = k A\), finding the numerical value of \(k\).

3 A circle has diameter \(d\), circumference \(C\), and area \(A\). Starting with the standard formulae for a circle, show that \(C d = k A\), finding the numerical value of \(k\).