For any chord through A, its length depends on its distance from O. The chord is shortest when this distance is greatest. This occurs when the chord is perpendicular to OA. Hence proved.
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As written, this statement appears inconsistent with the figure/text. Since each chord is perpendicular to diameter AB, its midpoint lies on AB. Therefore, the segment joining the two midpoints lies along AB, so it is parallel to AB, not perpendicular.
∠MOP is the angle subtended by arc MP at the centre, while ∠MNP is subtended by the same arc at the circumference. Therefore, the central angle is twice the angle at the circumference.
Half of the chords are 5 cm and 12 cm. Let the distance of the 24 cm chord from the centre be x. Then the other is x + 7. Solving gives x = 5 and radius = 13 cm.
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.