Question

Prove by vector method, the angle subtended on a semicircle is a right angle.

Hint:

If \( \mathbf{A} \cdot \mathbf{B} = 0 \), then the vectors are perpendicular.

Answer 1

Step 1: Define the Points 
Let the semicircle be centered at \( O \) with radius \( r \), and points \( A(-r,0) \), \( B(r,0) \), and \( P(x,y) \) on the semicircle. 
Step 2: Write the Vectors 
Vectors from \( A \) and \( B \) to \( P \): \[ \mathbf{AP} = (x + r, y), \quad \mathbf{BP} = (x - r, y). \] 
Step 3: Find the Dot Product 
\[ \mathbf{AP} \cdot \mathbf{BP} = (x + r, y) \cdot (x - r, y). \] \[ = (x + r)(x - r) + y^2. \] \[ = x^2 - r^2 + y^2. \] Using the equation of the semicircle: \[ x^2 + y^2 = r^2. \] \[ \mathbf{AP} \cdot \mathbf{BP} = r^2 - r^2 = 0. \] Since dot product is zero, \( AP \) is perpendicular to \( BP \), proving a right angle.