Circle II – Length of Chord & Perimeter of a Segment SS2 Further Mathematics Lesson Note

Length of chord and perimeter of a segment.

Consider a circle center O with radius r. If OC is the perpendicular distance from O to chord AB and angle AOB = 2θ, then the length of chord AB can be found as follows: In a right-angled triangle OCA

 AC    =    Sin  θ

  r

Cross multiply:

__

AC = r Sin θ

Since

AB  = 2  x AC

AB    =  2r  Sin θ

Where

r = radius of the circle

θ =Semi Vertical angle of the sector i.e. half of the angle subtended at the center by arc AB.

Also

The perimeter of segment ACBD    = Length of chord AB + length of arc ADB

= 2r Sin θ + θ   x 2 Пr

    360°

Example

1. In a circle of radius 6 cm, a chord is drawn 3 cm from the centre. 

(a) Calculate the angle subtended by the chord at the centre of the circle.

 (b) Find the length of the minor arc cut off by the chord

Solutions:

1. a) Let the required angle 

    = AOB = 2θ

Where

θ  = Semi-vertical angle of the sector.

 Then

Cos θ = 3 cm = 1

              6cm    2

 Cos θ = 0.5000

    θ = Cos-1  0.5000

    θ   = 60°

-: Required angle = 2θ

             = 2 x 60°

            = 120°

 b) Length of minor arc ADB =θ x 2 Пr

     1           2               360°

= 120 x 2 x 22 x 6cm

360       7

3

     1

= 4 x 22cm

       7

= 88cm = 12 4cm

     7               7

ASSIGNMENT 

1. Calculate the  area of a sector of a circle of radius 6cm which subtends an angle of 70° at the centre (π = 22/7)      A. 44cm2        B. 22cm2        C. 66cm2   D. 11cm2   E. 16.5cm2

2. What is the angle subtended at the centre of a sector of a circle of radius 2cm if the area of the sector is 2.2 cm2? (π = 22/7)A. 120°    B. 31 ½°   C. 43° D. 58°   E. 63°

3. What is the radius of a sector of a circle which subtends 140o at its centre and has an area of 99 m2?    A. 18m        27m        C 9m        E. 30m        E. 24m

4. A sector of 80° is removed from a circle of radius 12cm What area of the circle is left?     A. 253cm2   B. 704cm2C 176cm2D. 125cm2 E. 352cm2π

5. An arc of a circle radius 7 cm is 14cm long. What angle does the arc subtend at the centre of the circle?

6. An arc of a circle whose radius is 10 cm subtends an angle of 600 at the centre. Find the length of the arc.