Chapter 5: The circle

Circle :

In the adjoining figure,

O is the center of a circle.

Seg OP is the radius of a circle.

Seg AB is the diameter of a circle.

Seg CD is the chord of a circle.




Look at the alongside figure and answer the questions :

1) Write the name of the centre of a circle.

2) Write the name of a radius of a circle.

3) Write the name of the chord of a circle.

4) Write the name of the diameter of a circle.




Look at the alongside figure and write whether the 
following statements are true or false :

1) Seg TS is not a chord.                                        

2) Seg KM is a chord.

3) Seg CK is a radius.

4) Seg KM is not a diameter.

Property 1:

The perpendicular drawn from the centre to a

chord of the circle bisect the chord.

In the figure alongside, seg PT chord LM.

Using divider we can see, LT = TM.

Property 2:

Congruent chords in the same circle are at          

an equal distance from the centre of the circle.

In the figure alongside, chord AB is congruent
to chord CD,
seg OM chord AB,
seg ON chord CD.
Use the divider to see,
OM = ON.

Property 3:

Congruent chords in the same circle are form                                                                              

               congruent angles at the centre of the circle.

In the figure alongside, chord AB is congruent
to chord CD,

Use a protractor to measure, <AOB and <COD.
Then <AOB = <COD.
                                                                                                                                                                                                                                                                                                                                                                                  

Example 1.

In a circle, chord AB chord PQ.                                      

If chord AB is at a distance of 4 cm from the
center, what is the distance of the chord from
the centre?

Solution :

Chord AB is 4 cm from the centre of the circle.
Congruent chords are equidistant from the center.
So distance of the chord PQ from the center is 4 cm.

Example 2.

In a circle with center P, seg PT is perpendicular                                    

                     

to chord LM. l(LT) = 6.5 cm. Find the length of
seg TM.

Solution :

Seg PT is perpendicular to chord LM.
LT = 6.5 cm.
The perpendicular from the centre of a circle to a
chord bisect a chord.
So, LT = TM = 6.5 cm.
The length of the seg TM is 6.5 cm.

Example 3.

In a circle chord AB = chord CD and angle made                                

by the chord AB at the centre is 1200. What is the
measure of the angle made by the chord CD at the centre.

Solution :

The angle made by the chord AB at centre is 1200.
Congruent chords makes the congruent angles at the
centre of the circle.
So, The measure of the angle made by the chord CD
at the centre is 1200.

Example 4.

Find the length of the chord which is at a distance                          

of 6 cm from the center of a circle of a circle of
a radius 10 cm.

Solution :

Suppose C is the center of a circle.
KM is the chord at distance 6 cm from the center.
The radius of a circle KC is 10 cm.
Seg CK is perpendicular to chord KM.
Now in right angled triangle CLK,
By Pythagoras theorem,
(Hypotenuse)2 = (Base)2 + (Height)2
(10)2 = (6)2 + [l(KL)]2
100 - 36 = [l(KL)]2
64 = [l(KL)]2
8 = l(KL)
So, l(KM) = 2l(KL)
                  = 2 x 8 = 16 cm.
l(KM) = 16 cm.