Proved that the angle subtended on a semicircle is a right angle.

To demonstrate that the angle subtended at the circumference of a semicircle is a right angle, we can use a simple geometric approach.

Understanding the Setup

Consider a circle with center O and a diameter AB. Let C be any point on the circumference of the circle that lies on the semicircle formed by the diameter AB.

Visualizing the Triangle

Now, we can form triangle ABC, where:

• A and B are the endpoints of the diameter.
• C is any point on the semicircle.

Applying the Inscribed Angle Theorem

The Inscribed Angle Theorem states that an angle subtended by a diameter at the circumference of a circle is a right angle. In our case, angle ACB is subtended by the diameter AB.

Using Triangle Properties

In triangle ABC:

• AB is the diameter.
• By the theorem, angle ACB must equal 90 degrees.

Conclusion

This proves that any angle subtended at the circumference of a semicircle is indeed a right angle. Thus, angle ACB is a right angle, confirming the theorem.