Let A be any point within a given circle with centre O. Show that the shortest chord of the circle that passes Class 9

Let A be any point within a given circle with centre O. Show that the shortest chord of the circle that passes Class 9

Question 1.

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. Class 9

Solution:

Given: A is any point inside a circle of centre O.

To Show: Shortest chord of circle through A is perpendicular to OA.

Construction : Construct a chord CD such that CD is perpendicular to OA at point A. Construct another chord EF through A which is not perpendicular to OA. Draw OM perpendicular to chord EF.

Proof:

AOMA is a right angled triangle.

OA is hypotenuse of AOMA.

∴ OM must be shorter than OA.

Since distance of chord EF (i.e., OM) from centre is less than distance of chord CD (i.e, OA) from centre.

∴ Chord EF must be longer than chord CD.

∴ Shortest chord of the circle passing through A is the one that is perpendicular to OA.