Shaded region with arc

A question is this type if and only if it requires finding the area or perimeter of a shaded region bounded by straight sides and a circular arc, combining triangle and sector calculations.

7 questions · Standard +0.2

1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta
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Edexcel C2 2013 June Q8
10 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4f4eac7b-8908-480f-bb39-049944203fff-12_556_1392_210_283} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the design for a triangular garden \(A B C\) where \(A B = 7 \mathrm {~m} , A C = 13 \mathrm {~m}\) and \(B C = 10 \mathrm {~m}\). Given that angle \(B A C = \theta\) radians,
  1. show that, to 3 decimal places, \(\theta = 0.865\) The point \(D\) lies on \(A C\) such that \(B D\) is an arc of the circle centre \(A\), radius 7 m .
    The shaded region \(S\) is bounded by the arc \(B D\) and the lines \(B C\) and \(D C\). The shaded region \(S\) will be sown with grass seed, to make a lawned area. Given that 50 g of grass seed are needed for each square metre of lawn,
  2. find the amount of grass seed needed, giving your answer to the nearest 10 g .
AQA C2 Q4
Standard +0.3
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.
AQA C2 2006 January Q4
11 marks Standard +0.3
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.
AQA C2 2016 June Q6
11 marks Standard +0.3
6 The diagram shows a triangle \(A B C\). The lengths of \(A B , B C\) and \(A C\) are \(8 \mathrm {~cm} , 5 \mathrm {~cm}\) and 9 cm respectively.
Angle \(B A C\) is \(\theta\) radians.
  1. Show that \(\theta = 0.586\), correct to three significant figures.
  2. Find the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
  3. A circular sector, centre \(A\) and radius \(r \mathrm {~cm}\), is removed from triangle \(A B C\). The remaining shape is shown shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{e183578a-29a8-4112-b941-06c8894ed078-14_467_677_1462_685} Given that the area of the sector removed is equal to the area of the shaded shape, find the perimeter of the shaded shape. Give your answer in cm to three significant figures.
    [0pt] [6 marks]
Edexcel C2 Q6
9 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{288b99b5-1198-4463-baed-f0a4bf03e485-3_335_890_246_456} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows triangle \(A B C\) in which \(A C = 8 \mathrm {~cm}\) and \(\angle B A C = \angle B C A = 30 ^ { \circ }\).
  1. Find the area of triangle \(A B C\) in the form \(k \sqrt { 3 }\). The point \(M\) is the mid-point of \(A C\) and the points \(N\) and \(O\) lie on \(A B\) and \(B C\) such that \(M N\) and \(M O\) are arcs of circles with centres \(A\) and \(C\) respectively.
  2. Show that the area of the shaded region \(B N M O\) is \(\frac { 8 } { 3 } ( 2 \sqrt { 3 } - \pi ) \mathrm { cm } ^ { 2 }\).
Edexcel P1 2018 Specimen Q10
12 marks Moderate -0.3
\includegraphics{figure_4} The triangle \(XYZ\) in Figure 4 has \(XY = 6\) cm, \(YZ = 9\) cm, \(ZX = 4\) cm and angle \(ZXY = a\). The point \(W\) lies on the line \(XY\). The circular arc \(ZW\), in Figure 4, is a major arc of the circle with centre \(X\) and radius 4 cm.
  1. Show that, to 3 significant figures, \(a = 2.22\) radians. [2]
  2. Find the area, in cm\(^2\), of the major sector \(XZWX\). [3]
The region, shown shaded in Figure 4, is to be used as a design for a logo. Calculate
  1. the area of the logo [3]
  2. the perimeter of the logo. [4]
Edexcel C2 Q9
12 marks Standard +0.3
\includegraphics{figure_3} Fig. 3 Triangle ABC has AB = 9 cm, BC = 10 cm and CA = 5 cm. A circle, centre A and radius 3 cm, intersects AB and AC at P and Q respectively, as shown in Fig. 3.
  1. Show that, to 3 decimal places, ∠BAC = 1.504 radians. [3]
Calculate,
  1. the area, in cm², of the sector APQ, [2]
  2. the area, in cm², of the shaded region BPQC, [3]
  3. the perimeter, in cm, of the shaded region BPQC. [4]
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