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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)
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Solution
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
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