Math Labs with Activity – Line Drawn through Centre of a Circle to Bisect a Chord
To verify that the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord
- A sheet of white paper
- A sheet of tracing paper
- A geometry box
- A piece of cardboard
- A tube of glue
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).
Join OA, OB and OM.
In ΔAOM and BOM, we have
- OA = OB (each equal to r)
- OM = OM (common)
- 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.
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.
Step 6: Now, again fold the tracing paper such that . the point A lies exactly over the point B. What do you observe?
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.
It is verified that the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
From the above result it can be deducted that the perpendicular bisectors of two chords of a circle intersect at its centre.