Theorem: Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).
Consider a circle of centre O . It has PQ and RS two chords of equal length.
Now , draw a perpendicular from centre to PQ and RS intersecting them at L and M .
To Prove: OL = OM
Join OQ and OS.
Proof: We know that PQ =RS (given)
QL = `(PQ )/2` (perpendicular drawn from a centre to a chord bisect the chord)
Similarly, SM =`(RS)/2`
Therefore , QL = SM
Consider ∆OQL and ∆OSM
OQ = OS (radius of the same circle)
QL = SM (proved)
∠OLQ = ∠OMS = 90° (given )
∆OQL ≅ ∆OSM (SSA rule)
OL = OM (By CPCT).
Theorem: Chords equidistant from the centre of a circle are equal in length.
Shaalaa.com | Theorem: Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).
Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?
Find the length of a chord which is at a distance of 4 cm from the centre of the circle of radius 6 cm.
Find the length of a chord which is at a distance of 5 cm from the centre of a circle ofradius 10 cm.
Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.
If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.
If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (See given figure)