Thales’ Theorem states that if a triangle is inscribed in a circle where one of its sides is the diameter of the circle, then the angle opposite this side is a right angle (90 degrees).
Explanation
Consider a circle with a diameter AB and any point C on the circle (but not on AB). The triangle ABC is formed by connecting A, B, and C. Thales’ Theorem states that the angle ∠ACB is always 90 degrees, regardless of where C is placed on the circle.
Reasoning Behind the Theorem
The proof relies on properties of circles and inscribed angles:
- The center of the circle, O, is the midpoint of AB, meaning OA and OB are equal (radii of the circle).
- The two triangles OAC and OBC are isosceles.
- Using the sum of angles in a triangle and recognizing symmetrical relationships, we conclude that ∠ACB = 90°.
Applications
- Constructing right angles: If you need a precise 90-degree angle, drawing a semicircle and using Thales’ Theorem ensures accuracy.
- Tangent line construction: This theorem is used in the construction of tangents from an external point to a circle.
- Geometry proofs: It provides a fundamental way to show perpendicular relationships in circles.
Thales’ Theorem is a foundational result in Euclidean geometry and is frequently used in both theoretical and practical applications.
Note: Thales’ Theorem is pronounced as “THAY-leez” theorem (/ˈθeɪ.liːz/).
The name Thales comes from the ancient Greek mathematician Thales of Miletus, and its pronunciation follows the Greek-origin pronunciation rather than English phonetics.
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