Thales Theorem and Its Proof from "summary" of NCERT Class 10 Mathematics Solutions by JagranJosh
Thales Theorem states that if a triangle is drawn with its vertices on any three points on a circle, then the angle subtended by the triangle at the center of the circle is a right angle. This theorem is named after the Greek mathematician Thales of Miletus, who is considered one of the Seven Sages of Greece. To prove Thales Theorem, let us consider a circle with center O and three points A, B, and C on the circle such that A, B, and C form a triangle. Now, we draw lines OA, OB, and OC to connect the center O with the vertices of the triangle. Since OA, OB, and OC are radii of the circle, they are all equal in length. This implies that triangle OAB, triangle OBC, and triangle OCA are isosceles triangles. Now, we can observe that the angles subtended by these triangles at the center O are all equal because they are angles in the same segment of the circle. Let us denote these angles as ∠AOB, ∠BOC, and ∠COA. Since triangle OAB is isosceles, angle ∠OAB = ∠OBA. Similarly, in triangle OBC, angle ∠OBC = ∠OCB, and in triangle OCA, angle ∠OCA = ∠OAC. Adding these angles, we get: ∠OAB + ∠OBC + ∠OCA = ∠OBA + ∠BOC + ∠OAC Since ∠OAB = ∠OBA, ∠OBC = ∠BOC, and ∠OCA = ∠OAC, we can rewrite the equation as: ∠AOB + ∠BOC + ∠COA = ∠AOB + ∠BOC + ∠COA This simplifies to: ∠AOB + ∠BOC + ∠COA = ∠AOB + ∠BOC + ∠COA Thus, we have proved that the sum of the angles subtended by the triangle at the center of the circle is equal to itself, which means that the angle subtended is a right angle. This completes the proof of Thales Theorem.
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