Saturday, December 22, 2018

Area of Sector (CIE, IGCSE Edexcel)

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1 (CIE 2012, s, paper 12, question 8)

The figure shows a circle, centre $\D  O,$ with radius 10 cm. The lines $\D XA$ and $\D XB$ are tangents to the circle at $\D A$ and $\D B$ respectively, and angle $\D AOB$ is $\D \frac{2\pi}{3}$ radians.
(i) Find the perimeter of the shaded region. [3]
(ii) Find the area of the shaded region. [4]

2 (CIE 2012, s, paper 21, question 11)

The diagram shows a right-angled triangle $\D ABC$ and a sector $\D CBDC$ of a circle with centre $\D C$ and radius 12 cm. Angle $\D ACB = 1$ radian and $\D ACD$ is a straight line.
(i) Show that the length of $\D AB$ is approximately 10.1 cm. [1]
(ii) Find the perimeter of the shaded region. [5]
(iii) Find the area of the shaded region. [4]

3 (CIE 2012, w, paper 12, question 8)
The diagram shows an isosceles triangle $\D OBD$ in which $\D OB = OD = 18$ cm and angle $\D BOD = 1.5$ radians. An arc of the circle, centre $\D O$ and radius 10 cm, meets $\D OB$ at $\D A$ and $\D OD$ at $\D C.$
(i) Find the area of the shaded region. [3]
(ii) Find the perimeter of the shaded region. [4]

4 (CIE 2012, w, paper 13, question 9)
The diagram shows four straight lines, $\D AD, BC, AC$ and $\D BD.$ Lines $\D AC$ and $\D BD$ intersect at $\D O$ such that angle $\D AOB$ is $\D \frac{\pi}{6}$ radians. $\D AB$ is an arc of the circle, centre $\D O$ and radius 10 cm, and $\D CD$ is an arc of the circle, centre $\D O$ and radius 20 cm.
(i) Find the perimeter of $\D ABCD.$ [4]
(ii) Find the area of $\D ABCD.$ [4]

5 (CIE 2012, w, paper 21, question 8)

In the diagram $\D PQ$ and $\D RS$ are arcs of concentric circles with centre $\D O$ and angle $\D POQ = 1$ radian. The radius of the larger circle is $\D x$ cm and the radius of the smaller circle is $\D y$ cm.
(i) Given that the perimeter of the shaded region is 20 cm, express $\D y$ in terms of $\D x.$ [2]
(ii) Given that the area of the shaded region is 16cm$\D^2,$ express $\D y^2$ in terms of $\D x^2.$ [2]
(iii) Find the value of $\D x$ and of $\D y.$ [4]

6 (CIE 2013, s, paper 11, question 8)

The diagram shows a square $\D ABCD$ of side 16 cm. $\D M$ is the mid-point of $\D AB.$ The points $\D E$ and $\D F$ are on $\D AD$ and $\D BC$ respectively such that $\D AE = BF = 6$ cm. $\D EF$ is an arc of the circle centre $\D M,$ such that angle $\D EMF$ is $\D \theta $ radians.
(i) Show that $\D \theta = 1.855$ radians, correct to 3 decimal places. [2]
(ii) Calculate the perimeter of the shaded region. [4]
(iii) Calculate the area of the shaded region. [3]

7 (CIE 2013, s, paper 22, question 6)
The shaded region in the diagram is a segment of a circle with centre $\D O$ and radius $\D r$ cm. Angle $\D AOB = \frac{\pi}{3}$ radians.
(i) Show that the perimeter of the segment is $\D r\left(\frac{3+\pi}{3}\right).$ [2]
(ii) Given that the perimeter of the segment is 26 cm, find the value of $\D r$ and the area of the
segment. [5]

8 (CIE 2013, w, paper 13, question 8)
The diagram shows two concentric circles, centre $\D O,$ radii 4 cm and 6 cm. The points $\D A$ and $\D B$ lie on the larger circle and the points $\D C$ and $\D D$ lie on the smaller circle such that $\D ODA$ and $\D OCB$ are straight lines.
(i) Given that the area of triangle $\D OCD$ is 7.5 cm$\D ^2,$ show that $\D \theta = 1.215$ radians, to 3 decimal places. [2]
(ii) Find the perimeter of the shaded region. [4]
(iii) Find the area of the shaded region. [3]

9 (CIE 2013, w, paper 21, question 10)
The diagram shows a circle with centre $\D O$ and a chord $\D AB.$ The radius of the circle is 12 cm andangle AOB is 1.4 radians.
(i) Find the perimeter of the shaded region. [5]
(ii) Find the area of the shaded region. [4]

10 (CIE 2014, s, paper 12, question 7)
The diagram shows a circle, centre $\D O,$ radius 8 cm. Points $\D P$ and $\D Q$ lie on the circle such that the chord $\D PQ = 12$ cm and angle $\D POQ = \theta$ radians.
(i) Show that $\D \theta = 1.696,$ correct to 3 decimal places. [2]
(ii) Find the perimeter of the shaded region. [3]
(iii) Find the area of the shaded region. [3]

11 (CIE 2014, s, paper 23, question 1)

The diagram shows a sector of a circle of radius $\D r$ cm. The angle of the sector is 1.6 radians and the area of the sector is 500 cm$\D ^2 .$
(i) Find the value of $\D r.$ [2]
(ii) Hence find the perimeter of the sector. [2]

12 (CIE 2014, w, paper 13, question 6)
The diagram shows a sector, $\D AOB,$ of a circle centre $\D O,$ radius 12 cm. Angle $\D AOB = 0.9$ radians. The point $\D C$ lies on $\D OA$ such that $\D OC = CB.$
(i) Show that $\D OC = 9.65$ cm correct to 3 significant figures. [2]
(ii) Find the perimeter of the shaded region. [3]
(iii) Find the area of the shaded region. [3]

13 (CIE 2014, w, paper 21, question 11)
The diagram shows a sector $\D OPQ$ of a circle with centre $\D O$ and radius $\D x$ cm. Angle $\D POQ$ is 0.8 radians. The point $\D S$ lies on $\D OQ$ such that $\D OS = 5$ cm. The point $\D R$ lies on $\D OP$ such that angle $\D ORS$ is a right angle. Given that the area of triangle $\D ORS$ is one-fifth of the area of sector $\D OPQ,$ find
(i) the area of sector $\D OPQ$ in terms of $\D x$ and hence show that the value of $\D x$ is 8.837 correct to 4 significant figures, [5]
(ii) the perimeter of $\D PQSR,$ [3]
(iii) the area of $\D PQSR.$ [2]

Answers
1. (i) $\D 55.6 $
(ii) $\D 68.5$
2. (ii) $\D 54.3$
(iii) $\D 187$
3. (i) $\D 86.6$
(ii) $\D 55.5$
4. (i) $\D 73.9,$
(ii) $\D 231$
5. (i) $\D y = 3x - 20$
(ii) $\D y^2 = x^2 -32$
(iii) $\D x = 9; y = 7$
6. (ii) $\D P = 54.6$
(iii) $\D A = 115.25$
7. (ii) $\D r = 12.7;A = 14.6$
8. (ii) $\D 15.9 $
(iii) $\D 14.4$
9. (i) $\D 74.1 $
(ii) $\D 422$
10.  $\D P=48.7,A=178.5$
11.  $\D 25; 90$
12.  $\D P = 22.8;A = 19.4$
13.   (ii) $\D P = 19.8;A = 25$

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