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Arc length and Area of Sector (IB SL)

1 (IB/s1/2019/November/Paper2/q4)
[Maximum mark: 7]
The following diagram shows a right-angled triangle, ABC, with AC=10 cm,AB=6 cm and BC=8 cm.
The points D and F lie on [AC]. [BD] is perpendicular to [AC]. BEF is the arc of a circle, centred at A. The region R is bounded by [BD]. [DF] and are BEF.
(a) Find BẢ. [2]
(b) Find the area of R. [5]


2 (IB/sl/2019/May/paper2tz2/qu)
[Maximum mark: 7]
OAB is a sector of the circle with centre O and radius r, as shown in the following diagram.

The angle AOB is θ radians, where 0<θ<π2. The point C lies on OA and OA is perpendicular to BC.
(a) Show that OC=rcosθ. [1]
(b) Find the area of triangle OBC in terms of r and θ. [2]
(c) Given that the area of triangle OBC is 35 of the area of sector OAB, find θ. [4]


3 (IB/s1/2018/November/Paper1/q1)
[Maximum mark: 6]
The following diagram shows a circle with centre A and radius 6 cm.

The points B,C, and D lie on the circle, and BˆAC=2 radians.
(a) Find the area of the shaded sector. [2]
(b) Find the perimeter of the non-shaded sector ABDC. [4]


4 (IB/s1/2018/May/paper1tz2/q4)
[Maximum mark: 7]
The following diagram shows a circle with centre O and radius r cm.


The points A and B lie on the circumference of the circle, and AOB=θ. The area of the shaded sector AOB is 12 cm2 and the length of are AB is 6 cm. Find the value of r.


5 (IB/s1/2018/May/paper2tz1/q3)
[Maximum mark: 6]
The diagram shows a circle, centre O, with radius 4 cm. Points A and B lie on the circumference of the circle and AOOB=θ, where 0θπ.
(a) Find the area of the shaded region, in terms of θ.
(b) The area of the shaded region is 12 cm2. Find the value of θ. [3]


6 (IB/s1/2017/November/Paper1/q4)
[Maximum mark: 7 ]
The following diagram shows triangle ABC, with AB=3 cm,BC=8 cm, and ABC=π3.
(a) Show that AC=7 cm. [4]
(b) The shape in the following diagram is formed by adding a semicircle with diameter [AC] to the triangle.


Find the exact perimeter of this shape. [3]


7 (IB/s1/2017/May/paper2tz1/q5)
[Maximum mark: 7]
The following diagram shows the chord [AB] in a circle of radius 8 cm, where AB=12 cm.

Find the area of the shaded segment.


8 (IB/s1/2017/May/paper2tz2/q1)
[Maximum mark: 6]
The following diagram shows a circle with centre O and radius 40 cm.

The points A,B and C are on the circumference of the circle and AOC=1.9 radians.
(a) Find the length of arcABC.
(b) Find the perimeter of sector OABC.
(c) Find the area of sector OABC.


9 (IB/s1/2016/November/Paper2/q3)
[Maximum mark: 7]
The following diagram shows a circle, centre O and radius r mm. The circle is divided into five equal sectors.

One sector is OAB, and AOB=θ.
(a) Write down the exact value of θ in radians. [1]

The area of sector AOB is 20πmm2.
(b) Find the value of r. [3]
(c) Find AB. [3]


10 (IB/s1/2016/May/paper1tz2/q5)
[Maximum mark: 8]
The following diagram shows a triangle ABC and a sector BDC of a circle with centre B and radius 6 cm. The points A,B and D are on the same line.

AB=23 cm,BC=6 cm, area of triangle ABC=33 cm2,ABC is obtuse.
(a) Find ABC.
(b) Find the exact area of the sector BDC. [3]


11 (IB/s1/2015/November/Paper2/q1)
[Maximum mark: 6]
The following diagram shows a circle with centre O and radius 3 cm.

Points A,B, and C lie on the circle, and AOOC=1,3 radians.
(a) Find the length of arc ABC. [2]
(b) Find the area of the shaded region. [4]


12 (IB/s1/2015/May/paper1tz1/q2)
[Maximum mark: 5 ]
The following diagram shows a circle with centre O and a radius of 10 cm. Points A,B and C. lie on the circle.

Angle AOB is 1.2 radians.
(a) Find the length of are ACB.
(b) Find the perimeter of the shaded region.


13 (IB/sl/2015/May/paper2tz2/q10)
[Maximum mark: 16]
The following diagram shows a square ABCD, and a sector OAB of a circle centre O, radius r. Part of the square is shaded and labelled R.

AˆOB=θ, where 0.5θ<π.
(a) Show that the area of the square ABCD is 2r2(1cosθ).
(b) When θ=α, the area of the square ABCD is equal to the area of the sector OAB.

(i) Write down the area of the sector when θ=α.
(ii) Hence find α.
(c) When θ=β, the area of R is more than twice the area of the sector. Find all possible values of β.


Answer


[1](a) 0.927 (b) 8.05
[2](a) Show (b) A=12r2sinθcosθ (c) θ=.830
[3](a) Area =36 (b) Perimeter =12π
[4] r=4 cm
[5](a) A=8θ8sinθ (b) θ=2.27
[6](a) Show (b) 11+72π
[7] Area =22.5
[8](a) 76 (b) 156 (c) 1520
[9](a) θ=2π5 (b) r=10 (c) 11.8
[10](a) 5π6(150),π6(30) (b) 3π
[11](a) l=3.9 (b) area =9π5.85
[12](a) 12 (b) P=8+20π
[13](a) Show (b) (i) 12αr2 (ii) α=0.511 (c) 1.31<β<2.67

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