JAMB 1987 · UME · Q45

What is the locus of the mid-points of all chords of length 6 cm within a circle of radius 5 cm and centre $O$ ?

Worked solution (try it first)

The line from the centre to the mid-point of a chord is perpendicular to the chord.
Half of the 6 cm chord is 3 cm.
By Pythagoras, with the radius 5 cm as hypotenuse, the distance from $O$ to the mid-point is $\sqrt{5^2 - 3^2} = \sqrt{16} = 4$ cm.
Every such mid-point is 4 cm from $O$ , so the locus is a circle of radius 4 cm and centre $O$ , option A.