Question

Prove that A(−5, 4), B(−1, −2), C(5, 2) are the vertices of an isosceles right-angled triangle.

Answer 1

Step 1: Find the lengths of the sides using the distance formula

The distance formula between two points (x₁, y₁) and (x₂​, y₂​) is:

`"Distance" = sqrt((x_2 − x_1)^2 + (y_2 − y_1)^2)`

For AB:

AB = `sqrt((−1 − (−5))^2 + (−2 − 4)^2) = sqrt((4)^2 + (−6)^2) = sqrt(16 + 36) = sqrt52`

For BC: 

BC = `sqrt((5 + 1)^2 + (2 + 2)^2) = sqrt((6)^2 + (4)^2) = sqrt(36 + 16) = sqrt52`

For CA: 

CA = `sqrt((5 + 5)^2 + (2 − 4)^2) = sqrt((10)^2 + (−2)^2) = sqrt(100 + 4) = sqrt104`

so AB = BC = `sqrt52`, triangle ABC is isosceles.

= AB² + BC² = AC²   ...[Apply Pythagoras Theorem]

= 52 + 52 = AC²

AC² = 104

Triangle ABC is an isosceles right-angled triangle.