Question 16 - End-of-Chapter Exercises - Chapter 5 Class 9 - I’m Up and Down, and Round and Round (Ganita Manja

Transcript

Question 16 Consider all chords of a circle of a fixed length. What is the shape formed by the midpoints of all these chords? Let’s do this step-by-step Step 1 of 6 The Single Chord Let's start with an outer circle of radius . We draw a single chord of a fixed length . The green dot marks the exact midpoint of this chord. Previous Next StepStep 2 of 6 2. Multiple Chords Now, imagine drawing many different chords around the circle, all with the exact same length . We mark the midpoint of each one. Notice a pattern starting to emerge? Previous Next StepStep 3 of 6 3. The Locus (The Answer) If we draw every possible chord of length L, their midpoints form a continuous path. Answer: The midpoints arrange themselves perfectly into a smaller, concentric circle! Previous Next StepStep 4 of 6 4. Proof: The Perpendicular To prove this mathematically, draw a line from the center to the midpoint M. A fundamental circle theorem states that a line drawn from the center to bisect a chord is always perpendicular ( ) to that chord. Previous Next StepStep 5 of 6 5. Proof: The Right Triangle Next, draw the radius from the center to the edge of the chord. This forms a right-angled triangle! We know the hypotenuse is , and the base is exactly half the chord's length, L/2. Previous Next StepStep 6 of 6 6. Proof: Pythagoras Let's call the distance to the midpoint r. By Pythagoras' theorem: Solving for gives: . Since and are fixed constant numbers, the distance is always constant! A path with a constant distance from a center point is, by definition, a perfect circle. Previous Next Step