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Math Labs with Activity - Equal Chords of Congruent Circles are Equidistant

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Math Labs with Activity – Equal Chords of Congruent Circles are Equidistant

OBJECTIVE

To verify that equal chords of congruent circles are equidistant from their corresponding centres.

Materials Required

  1. A sheet of white paper
  2. A piece of cardboard
  3. A sheet of tracing paper
  4. A tube of glue
  5. A geometry box

Theory
The theorem to be verified is an extension of the theorem verified in Activity 23. Here, instead of a single circle, two congruent circles are involved.

Procedure
Step 1: Paste the sheet of white paper on the cardboard.
Mark two points O and O’ on this paper. With O as centre, draw a circle of any radius.
With O’ as centre, draw a circle of the same radius. (Note that the two circles must not intersect each other.) Thus, we have two congruent circles.
Step 2: Place the needle point of a pair of compasses at any point A on the first circle and taking any radius cut the circle, say, at a point B.
Again, place the needle point of the compasses at any point P on the second circle and taking the same radius cut the circle, say, at a point Q. Join AB and PQ. Then, AB and PQ are equal chords of congruent circles.
Step 3: Trace the two circles along with their chords AB and PQ on the tracing paper.
Step 4: Fold the tracing paper along the line that passes through the centre O of the first circle and cuts the chord AB such that the part of the chord AB that lies on one side of the line of fold overlaps the part on the other side.
Make a crease and unfold the paper. Mark the point M where the line of fold cuts the chord AB. Join OM. Then, OM is the perpendicular bisector of the chord AB.
Therefore, the distance of the chord AB from the centre O of the circle is equal to OM [see Figure 24.1(a)].
Math Labs with Activity - Equal Chords of Congruent Circles are Equidistant 1
Step 5: Again, fold the tracing paper along the line that passes through the centre O’ of the second circle and cuts the chord PQ such that the part of the chord PQ that lies on one side of the line of fold overlaps the part on the other side. Make a crease and unfold the paper. Mark the point N where the line of fold cuts the chord PQ. JoinO ‘N. Then, O’N is the perpendicular bisector of the chord PQ. Therefore, the distance of the chord PQ from the centre O’ of the circle is equal to O’N [see Figure 24.1(b)].
Math Labs with Activity - Equal Chords of Congruent Circles are Equidistant 2
Step 6: Cut the two circles from the tracing paper. Place ‘ the second circle over the first circle such that O’N overlaps OM.

Observations
When Figure 24.1(b) is superimposed upon Figure 24.1(a), we observe that O’N exactly covers OM. Therefore, OM =O’N.

Result
It is verified that equal chords of congruent circles are equidistant from their corresponding centres.

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