Mathematics
If a diameter of a circle bisects each of the two chords of a circle, then prove that the chords are parallel.
Answer
Let diameter POQ bisect chords AB and CD at L and M respectively.
Then, OL ⟂ AB and OM ⟂ CD
∴ ∠ALO = ∠OMD = 90°
These two angles are alternate interior angles, where PQ is transversal intersecting chords AB and CD.
Since, these angles are equal
∴ AB ∥ CD.
Hence, the two chords are parallel.
Related Questions
In the adjoining figure, AB is a chord of a circle with centre O and BC is a diameter. If OD ⟂ AB, show that CA = 2OD and CA ∥ OD.
In the adjoining figure, P is a point of intersection of two circles with centres C and D. If the straight line APB is parallel to CD, prove that AB = 2CD.
If two chords of a circle are equally inclined to the diameter through their point of intersection, prove that the chords are equal.
Show that equal chords of a circle subtend equal angles at the centre of the circle.