Angle in a Semicircle
Explore how any angle inscribed in a semicircle is always a right angle (90°). Move point C around the semicircle and observe how the angle at C always remains 90°.
No matter where you place point C on the semicircle, the angle ACB is always 90 degrees.
The Mathematics Behind the Theorem
Thales' Theorem states that if A, B, and C are points on a circle where AC is the diameter, then the angle ABC is a right angle.
- Central Angle: An angle whose vertex is at the center of the circle
- Inscribed Angle: An angle whose vertex is on the circle
- Semicircle: Half of a circle, bounded by a diameter
1. Draw radius OC to create two isosceles triangles (AOC and BOC)
2. In triangle AOC, angles OAC and OCA are equal (α)
3. In triangle BOC, angles OBC and OCB are equal (β)
4. Since AOB is a straight line, 2α + 2β = 180°
5. Therefore α + β = 90°, making angle ACB a right angle
