Class 9 Maths Circles Equal chords & their distance from center

Equal chords & their distance from center

In the given figure, we have to find distance of point P from line AB. We can draw multiple lines from point P to line AB, such as PC, PD, PE, PF, PG ,PA, PB. All these lines are of different length. So, we consider. the length of the perpendicular PE from a point P to a line AB as the distance of the line from the point.

Also notice that the length of the perpendicular PE from a point P to a line AB is the least distance. That is PE is smallest of PA, PB, PC, PD, PE, PF, PG etc.

If point P lines on Line AB, then the perpendicular distance is 0.

 

Length of chord & distance from center: A circle can have infinitely many chords. In the figure given we observe that longer chord is nearer to the centre than the smaller chord. Diameter is the longest chord from the centre, since the centre lies on it, the distance is zero.  AB is the longest chord in the figure shown & IJ is the shortest chord. Also observe that distance of AB is shortest from center & that of IJ is largest.

Theorem 6: Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centers).

In the figure, if Chord AB= CD, then it is observed that perpendiculars OP = OQ

Refer ExamFear video lessons for Proof of this theorem

Converse Theorem 7: Chords equidistant from the centre of a circle are equal in length.

In the figure, if perpendiculars OP = OQ, then it is observed that chord AB= CD.

Refer ExamFear video lessons for Proof of this theorem

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