If a line segment joining mid-points of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel.
Solution:
Consider a circle with centre O
AB and CD are two chords of a circle
PQ is the diameter which bisects the chord AB and CD at L and M.
The diameter PQ passes through the centre O of the circle.
We have to prove that the chords are parallel.
Since L is the midpoint of AB
OL ⊥ AB
We know that the line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
∠ALO = 90° -------------------------- (1)
Similarly, OM ⊥ CD
∠OMD = 90° ---------------------- (2)
From (1) and (2),
∠ALO = ∠OMD = 90°
We know that the alternate angles are equal.
From the figure,
∠ALO and ∠OMD are alternate angles.
Therefore, AB || CD
✦ Try This: Suppose two chords of a circle are equidistant from the centre of the circle. Prove that the chords have equal length.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 10
NCERT Exemplar Class 9 Maths Exercise 10.3 Problem 5
If a line segment joining mid-points of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel
Summary:
If a line segment joining mid-points of two chords of a circle passes through the centre of the circle, it is proven that the two chords are parallel
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