Geometric Proof: Diameter Bisecting a Chord is Perpendicular

Geometric Proof of Perpendicularity

• 💡 The video aims to prove that a diameter bisecting a chord is always perpendicular to that chord.
• 🎯 We start with a circle, a diameter, and a chord that the diameter divides into two equal segments.

Constructing Congruent Triangles

• 📐 To prove perpendicularity, we construct two triangles by drawing radii from the center of the circle to the endpoints of the chord.
• 📏 Since all radii of a circle are equal in length, two sides of the constructed triangles are equal.
• 🤝 The shared side (the diameter segment) and the two equal radii allow us to establish that the two triangles are congruent by Side-Side-Side (SSS).

Angle Relationships and Conclusion

• 🌟 Congruent triangles mean their corresponding angles are equal. The angles formed at the intersection of the diameter and the chord are therefore equal.
• 📐 Let these equal angles be represented by ( \theta ) and ( \phi ). We know ( \theta = \phi ).
• 🔄 These two angles are also supplementary because they form a straight line along the chord, meaning ( \theta + \phi = 180^\circ ).
• 💡 Substituting ( \theta ) for ( \phi ) (since they are equal), we get ( 2\theta = 180^\circ ), which means ( \theta = 90^\circ ).
• ✅ Therefore, both angles are 90 degrees, proving that the diameter is perpendicular to the chord it bisects.