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.
Let A be any point within a given circle with centre O. Show that the shortest chord of the circle that passes through point A is the one that is perpendicular to OA.
Share
Let a chord through A meet the circle at B and C. The perpendicular distance of this chord from O determines its length. Since A lies inside the circle, the greatest possible distance of a chord through A from O is OA itself. This occurs when the chord is perpendicular to OA. A chord farther from the centre is shorter. Therefore, among all chords passing through A, the shortest chord is the one perpendicular to OA. Hence proved.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 5 I’m Up and Down and Round and Round (2026-27):
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-5/