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Math Labs with Activity - Line Drawn through Centre of a Circle to Bisect a Chord

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Math Labs with Activity – Line Drawn through Centre of a Circle to Bisect a Chord

OBJECTIVE

To verify that the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord

Materials Required

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

Theory
The theorem to be verified is the converse of the theorem verified in Activity 18.
The theorem can be proved as below.
Consider a circle with centre O and radius r having a chord AB. Let M be the midpoint of the chord AB (see Figure 19.1).
Math Labs with Activity – Line Drawn through Centre of a Circle to Bisect a Chord 1
Join OA, OB and OM.
In ΔAOM and BOM, we have

  1. OA = OB (each equal to r)
  2. OM = OM (common)
  3. AM = MB (M being the midpoint of AB)

Therefore, ΔAOM is congruent to ΔBOM (by SSS-criterion).
So, ∠OMA = ∠OMB – 90° (since ∠OMA and ∠OMB form a linear pair) i.e., OM ⊥ AB.

Procedure
Step 1: Paste the sheet of white paper on the cardboard and mark a point O on this paper. With O as the centre, draw a circle with any radius.
Step 2: Draw a chord AB in this circle.
Step 3: Trace the circle along with the chord AB on the tracing paper.
Step 4: Fold the tracing paper along a line which cuts the chord AB in such a way that the part of the chord that lies on one side of this line overlaps the part on the other side and the point A lies exactly over the point B.
Form a crease and unfold the tracing paper. Mark the point M where the line of fold meets the chord AB. Then, M is the midpoint of the chord AB.
Step 5: Join OM as shown in Figure 19.2. Then, OM is the line drawn through the centre of the circle to bisect the chord.
Math Labs with Activity – Line Drawn through Centre of a Circle to Bisect a Chord 2
Step 6: Now, again fold the tracing paper such that . the point A lies exactly over the point B. What do you observe?

Observations
We observe that the fold is along the line OM. This shows that OM is perpendicular to AB. Thus, OM is the perpendicular bisector of the chord AB.

Result
It is verified that the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

Remarks:
From the above result it can be deducted that the perpendicular bisectors of two chords of a circle intersect at its centre.

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