Question

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

Answer 1

Let AB be the diameter of the circle with centre ‘O’

Let P be any point on the semi-circle.

To prove ∠APB = 90°

We have OA = OB = OP  .......(radii)

Now `bar"PA" - bar"PO" + bar"OA"`

`bar"PB" = bar"PO" + bar"OB"`

= `bar"PO" - bar"OA"`  .....`("Since"  bar"OB" = - bar"OA")`

∴ `bar"PA" * bar"PB" = (bar"PO" + bar"OA") * (bar"PO" - bar"OA")`

= `(bar"PO")^2 - (bar"OA")^2`

= (PO)² – (OA)²

`bar"PA" * bar"PB"` = 0

∴ `bar"PA"` ⊥' to `bar"PB"`

This gives ∠APB = 90°.

Hence the result.