TR-07Z: First Proof Thales' Theorem (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial aims to provide a comprehensive understanding of Thales' Theorem, using basic trigonometric concepts. Thales' Theorem states that if A, B, and C are points on a circle where line segment AC is the diameter, then angle ABC is a right angle. This theorem is fundamental in geometry and trigonometry and is applicable in various mathematical and real-world problems.

Step 1: Understand the Components of Thales' Theorem

To effectively prove Thales' Theorem, familiarize yourself with the following components:

• Circle: A round shape where every point is equidistant from the center.
• Diameter: A line segment passing through the center of the circle, connecting two points on the circumference.
• Inscribed Angle: An angle formed by two chords in a circle which share an endpoint.

Practical Advice

• Draw a circle and label the center O, diameter endpoints A and C, and a point B on the circumference.
• Visualize how the angle ABC is formed with respect to the diameter AC.

Step 2: Set Up the Proof Using Trigonometry

We will use basic trigonometric functions to prove that angle ABC is a right angle.

1. Identify Coordinates:

   • Place point A at (-1, 0) and point C at (1, 0) on a Cartesian plane.
   • Let point B be at (x, y), where B lies on the circle defined by the equation ( x^2 + y^2 = 1 ).

2. Apply Trigonometric Concepts:

   • The coordinates of points A, B, and C can be used to determine the slopes of the lines AB and BC.

Slope Calculation

• Slope of AB: [ m_{AB} = \frac{y - 0}{x - (-1)} = \frac{y}{x + 1} ]

• Slope of BC: [ m_{BC} = \frac{y - 0}{x - 1} = \frac{y}{x - 1} ]

3. Find Relationship Between Slopes:
   • For angle ABC to be a right angle, the product of the slopes must equal -1: [ m_{AB} \cdot m_{BC} = -1 ]

Practical Tip

• Set up a scenario with specific values for x and y to verify that the condition holds true.

Step 3: Conclude the Proof

Using the relationship derived from the slopes, confirm that:

• If ( x^2 + y^2 = 1 ), then the slopes will satisfy the condition for perpendicular lines.

Final Verification

• Substitute ( x = \cos(\theta) ) and ( y = \sin(\theta) ) into the slope equations to demonstrate that the theorem holds for any angle θ.

Conclusion

Thales' Theorem is a powerful concept in geometry, proven through basic trigonometric principles. By understanding the components and applying trigonometric functions, you can confidently demonstrate that the angle formed by a diameter and a point on the circle is a right angle. As a next step, explore more complex applications of this theorem in real-world scenarios or delve deeper into advanced trigonometric proofs and their implications in geometry.