Question

Show that the given points form a right angled triangle and check whether they satisfy Pythagoras theorem

A(1, – 4), B(2, – 3) and C(4, – 7)

Answer 1

The vertices are A(1, – 4), B(2, – 3) and C(4, – 7)

Slope of a line = `(y_2 - y_1)/(x_2 - x_1)`

Slope of AB = `(-3 + 4)/(2 - 1) = 1/1` = 1

Slope of BC = `(-7 + 3)/(4 -2) = (-4)/2` = – 2

Slope of AC = `(-7 + 4)/(4 - 1) = - 3/3` = – 1

Slope of AB × Slope of AC = 1 × – 1 = – 1

∴ AB is ⊥^r to AC

∠A = 90°

∴ ABC is a right angle triangle

Verification:

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

AB = `sqrt((2 - 1)^2 + (-3 + 4)^2`

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

= `sqrt(2)`

BC = `sqrt((4 - 2)^2 + (- 7 + 3)^2`

= `sqrt((2)^2 + (- 4)^2`

= `sqrt(4 + 16)`

= `sqrt(20)`

AC = `sqrt((4 - 1)^2 + (-7 + 4)^2`

= `sqrt(3^2 + (- 3)^2`

= `sqrt(9 + 9)`

= `sqrt(18)`

BC² = AB² + AC²

`(sqrt(20))^2 = (sqrt(2))^2 + (sqrt(18))^2`

20 = 2 + 18

20 = 20

⇒ Pythagoras theorem verified