Equal chords of a circle are equidistant from the centre

   

Topic: Equal chords of a circle are equidistant from the centre

Objective: To prove that equal chords of a circle are equidistant from the centre of the circle.

Pre-requisite knowledge:

(1) Construction of equal chords of a circle.

(ii) To draw a line perpendicular to a line from a point outside it (by paper folding).

Materials required:

(1) Tracing papers

(ii) Geometry box

(iii) Coloured ball point pens,

To perform the activity: Steps;

1. Draw a circle of any radius with centre O on a tracing paper

in as shown fig. 6.1.

2. Draw two equal chords AB and CD of the circle (you may use compasses) as shown in fig. 6.1.

3. From O, draw OM 1 AB and ON 1 CD (by paper folding).

4. Fold the figure/paper so that A falls on C and B falls on D (this is possible because AB = CD) and form a crease as shown dotted in fig. 6.2

Result:

We observe that OM exactly covers ON

OM = ON

the distance of O from AB = the distance of O from CD. Hence equal chords of a circle are equidistant from the centre of the circle.

Hote: The above result is true in case of congruent circles also be equal chords of congruent circles are equidistant from their respective centres.

To see this :

1. Draw two contruent circles (i.e. circles of equal radii) with centre O and O on tracing papers as shown in fig. 6.3.

2. Draw equal chords AB and CD of the two circles as shown in fig. 6.3.

3. From O, draw OM LAB and from O', draw ON 1 CD (by paper folding

4. Superimpose the first circle on the second circle such that O falls O and A falls on C (this is possible because circles have equal radius i OA = O'C) as shown in fig. 6.4.

Result:

We observe that B falls on D and OM exactly covers O'N

OM = ON

the distance of O from AB = the distance of O' from CD.

Hence equal chords of congruent circles are equidistant from their respective centres.

Thus we have proved:

Equal chords of a circle (or in congruent circles) are equidistant from the centre (or centres).