Question

If A(a, 0), B(1, 1) and C(0, b) form a triangle, right angled at B when joined, then establish a relation between a and b.

Answer 1

By distance formula,

`AB = sqrt((a - 1)^2 + (0 - 1)^2`

= `sqrt(a^2 + 1 - 2a + 1)`

= `sqrt(a^2 - 2a + 2)`

`BC = sqrt((0 - 1)^2 + (b - 1)^2`

= `sqrt(1 + b^2 + 1 - 2b)`

= `sqrt(b^2 - 2b + 2)`

`AC = sqrt((0 - a)^2 + (b - 0)^2`

= `sqrt(a^2 + b^2)`

Since, ΔABC is a right angled triangle.

By Pythagoras theorem,

AB² + BC² = AC²

⇒ a² – 2a + 2 + b² – 2b + 2 = a² + b²

⇒ –2a – 2b + 4 = 0

⇒ 2a + 2b = 4

⇒ a + b = 2

Hence, the required relation between a and b is established.