Theorem : 3 Angle in a Semicircle is a Right Angle

Objective

• Understand the geometric property of angles formed in a semicircle
• Use the theorem to solve problems involving right-angled triangles in circles
• Apply this theorem in proofs and constructions involving circles

Definition / Concept

In any triangle inscribed in a semicircle, if one side of the triangle is the diameter of the circle, then the angle opposite the diameter is always a right angle.

This is known as Thales’ Theorem.

$$\text{If } AB \text{ is the diameter and } C \text{ is any point on the semicircle, then} \quad \boxed{\angle ACB = 90^\circ}$$If AB is the diameter and C is any point on the semicircle, then∠ACB=90∘​

This holds true regardless of where point \(C\) lies on the arc of the semicircle.

Question

Question