Mathematics
If two chords of a circle are equally inclined to the diameter through their point of intersection, prove that the chords are equal.
Answer
Let AB and AC be two chords.
AOD be a diameter such that ∠BAO = ∠CAO.
OL ⟂ AB and OM ⟂ AC.
In △OLA and △OMA,
OA = OA [common side]
∠OLA = ∠OMA = 90°
∠LAO = ∠MAO [AO bisects ∠A]
∴ △OLA ≅ △OMA [By A.A.S. rule]
Then,
OL = OM (By C.P.C.T.C.)
AB = AC [Chords which are equidistant from centre are equal]
Hence, proved that the two chords are equal.
Related Questions
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 a diameter of a circle bisects each of the two chords of a circle, then prove that the chords are parallel.
Show that equal chords of a circle subtend equal angles at the centre of the circle.
In the given figure, equal chords AB and CD of a circle with centre O cut at right angles at P. If L and M are mid-points of AB and CD respectively, prove that OLPM is a square.
