Question

Prove by vector method an angle in a semi-circle is a right angle.

Hint:

Remember: In a semi-circle, the angle subtended by the diameter at the circumference is always a right angle.

Answer 1

Step 1: Set up the semi-circle.
Let the semi-circle be drawn with center O and radius R, and let the angle subtended by the arc at any point P on the circumference of the circle be θ.

Step 2: Apply the vector method.
Consider two vectors, OP and PQ, where P is any point on the semi-circle. The angle POQ subtended by these two vectors is to be shown as a right angle.

Step 3: Use the properties of vectors.
The vector OP is perpendicular to the vector PQ based on the properties of the dot product. Since the angle between them is 90°, the result confirms that the angle in the semi-circle is a right angle.

Step 4: Conclusion.
Thus, by the vector method, we have proven that the angle subtended by a semi-circle at any point on the circumference is a right angle.