Consider all chords of a circle of a fixed length. What is the shape formed by the midpoints of all these chords?
The shape formed by the midpoints of all these chords is a circle.
Explanation
According to the textbook context, equal chords are equidistant from the centre. Since all chords have a fixed length, they are equal and thus equidistant from the centre. The perpendicular from the centre to a chord bisects it, meaning this perpendicular distance is exactly the distance from the centre to the midpoint of the chord. Because this distance is constant for all such chords, the midpoints all lie at a fixed distance from the centre, forming a circle.
Solution Steps
Step 1: Since all chords are of a fixed length, they are equal chords. Equal chords are equidistant from the centre.
Step 2: A perpendicular drawn from the centre to a chord bisects it, so this perpendicular passes through the midpoint of the chord.
Step 3: The distance from the centre to the midpoint of each chord is equal to the fixed distance from the centre to the chord.
Step 4: All points at a fixed distance from the centre form a circle. Hence, the midpoints form a circle.