Question

Prove that the points A (1, -3), B (-3, 0) and C (4, 1) are the vertices of an isosceles right-angled triangle. Find the area of the triangle.

Answer 1

AB =`sqrt((-3 -1)^2 + (0 +3)^2) = sqrt(16+9) = sqrt(25)` = 5

BC =`sqrt((4 + 3)^2 + (1 +0)^2)= sqrt(49+1)= sqrt(50) = 5sqrt(2)`

CA =`sqrt((1 -4)^2 + (-3 - 1)^2) = sqrt(9 + 16) = sqrt(25)` = 5

∵ AB = CA

A, B, C are the vertices of an isosceless triangle.

AB² + CA²  = 25 + 25 = 50

BC² = `(5sqrt(2))^2` = 50

∴ AB² + CA² = BC²

Hence, A, B, C are the vertices of a right-angled triangle.

Hence, ΔABC is an isosceles right-angled triangle.

Area of ΔABC = `(1)/(2) xx "AB" xx "CA"`

= `(1)/(2) xx 5 xx 5`

= 12.5 sq.units