Question 25 - End-of-Chapter Exercises - Chapter 5 Class 9 - I’m Up and Down, and Round and Round (Ganita Manja

Transcript

Question 25 In a circle, two chords CC^′ and DD ' are drawn perpendicular to a diameter AB. Prove that the segment MM ' joining the midpoints of the chords CD and C^′ D^′ is perpendicular to AB. We use the concept of reflectional symmetry across the diameter to prove this. Step 1 of 5 The Setup We start with a circle and a horizontal diameter AB. We draw two chords, CC' and DD', such that they are exactly perpendicular ( ) to the diameter AB . Previous Next StepStep 2 of 5 2. The Line of Symmetry A fundamental circle theorem states that a diameter perpendicular to a chord perfectly bisects it. This means the top half of each chord equals the bottom half. Because of this, the diameter AB acts as a perfect mirror (line of symmetry) for the entire figure. The top half of the circle is an exact reflection of the bottom. Previous Next StepStep 3 of 5 3. The Connecting Chords Now we draw the chord CD on the top, and C'D' on the bottom. Because point C is a reflection of , and is a reflection of , the entire line segment CD is a perfect mirror reflection of C'D' across the diameter AB . Previous Next StepStep 4 of 5 4. The Midpoints Let's find the midpoint of the top chord CD , and the midpoint of the bottom chord C'D'. Since the two lines are perfect reflections of each other, their exact centers must also be perfect reflections of each other! Previous Next StepStep 5 of 5 5. The Final Conclusion Finally, we draw a segment joining and . By definition of reflectional symmetry, the line connecting any point to its mirror image must cross the mirror line at exactly a right angle. Conclusion: Since and are mirror reflections across the diameter AB , the segment is necessarily perpendicular to AB!