In a rectangle, the diagonals are equal and bisect each other. Since the rectangle is cyclic, each diagonal is a diameter of the circle. Therefore, their intersection is the midpoint of a diameter, which is the centre.
Show that if a rectangle is inscribed in a circle, then the point of intersection of its diagonals must lie at the centre of the circle.
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Let ABCD be a rectangle inscribed in a circle. Its diagonals AC and BD are equal and bisect each other. Since AC and BD are equal chords passing through the centre, they are diameters of the circle. The diagonals of the rectangle intersect at their common midpoint. The centre of a circle is the midpoint of every diameter. Therefore, the point where the diagonals of the rectangle intersect must be the centre of the circle.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 5 I’m Up and Down and Round and Round (2026-27):
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but we have to prove that it is passing through the centre so how can we tell that the diagonals are the diameter
Let ABCD be a rectangle inscribed in a circle, and let its diagonals AC and BD intersect at O.
1. Since ABCD is a rectangle, its diagonals are equal and bisect each other.
Therefore,
AC = BD and AO = OC, BO = OD.
2. Since AC and BD join two points on the circle, they are chords of the circle.
3. The diagonals of a rectangle are equal, so AC and BD are equal chords of the circle.
4. We know that equal chords of a circle are equidistant from the centre.
5. Also, since the diagonals bisect each other at O, O is the midpoint of both chords AC and BD.
6. The perpendicular from the centre of a circle to a chord bisects the chord. Therefore, the perpendicular bisectors of chords AC and BD both pass through the centre.
7. But O is the midpoint of both chords, so the perpendicular bisectors meet at O. Hence, O is the centre of the circle.
Therefore, the point of intersection of the diagonals of a rectangle inscribed in a circle is the centre of the circle.
Hence proved.