Mathematics
A straight line is drawn cutting two equal circles and passing through the mid-point M of the line joining their centers O and O'.
Prove that the chords AB and CD, which are intercepted by the two circles, are equal.
Circles
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Answer
Draw OP ⊥ AB and O'Q ⊥ CD.
In △ OMP and △ O'MQ,
⇒ ∠OMP = ∠O'MQ (Vertically opposite angles are equal)
⇒ ∠OPM = ∠O'QM (Both equal to 90°)
⇒ OM = O'M (As, M is the mid-point of OO')
∴ △ OMP ≅ △ O'MQ (By A.A.S. axiom)
We know that,
Corresponding parts of congruent triangle are equal.
∴ OP = O'Q.
We know that,
Two chords of a circle or equal circles which are equidistant from the center are equal.
∴ AB = CD.
Hence, proved that AB = CD.
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Related Questions
Two parallel chords are drawn in a circle of diameter 30.0 cm. The length of one chord is 24.0 cm and the distance between the two chords is 21.0 cm; find the length of the other chord.
A chord CD of a circle, whose center is O, is bisected at P by a diameter AB.
Given OA = OB = 15 cm and OP = 9 cm. Calculate the lengths of :
(i) CD
(ii) AD
(iii) CB.
M and N are the mid-points of two equal chords AB and CD respectively of a circle with center O. Prove that :
(i) ∠BMN = ∠DNM
(ii) ∠AMN = ∠CNM.
Two equal chords AB and CD of a circle with center O, intersect each other at point P inside the circle. Prove that :
(i) AP = CP
(ii) BP = DP