Consider all chords of a circle of a fixed length. What is the shape formed by the midpoints Class 9

Consider all chords of a circle of a fixed length. What is the shape formed by the midpoints Class 9

Question 1.

Consider all chords of a circle of a fixed length. What is the shape formed by the midpoints of all these chords? Class 9

Solution:

Consider a circle of centre O and radius r.

All chords have the same fixed length l ...[Given]

Since equal chords are equidistant from the centre.

∴ Every chord of length l is at the same distance d = $\sqrt{\left(r^2-\left(\frac{l}{2}\right)^2\right)}$ from O.

... [Baudhayana-Pythagoras theorem]

∴ The midpoint of each chord is at distance d from O ... [perpendicular from O to chord]

∴ All midpoints lie at a constant distance d from O.

∴ The locus of midpoints of all equal chords is a circle with centre O and radius d.

∴ The midpoints of all equal chords form a circle concentric with the given circle, with radius d = $\sqrt{\left(r^2-\frac{l^2}{4}\right)}$