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Question
Find the lengths of the sides of the triangle and also determine the type of a triangle:
A(2, -1, 0), B(4, 1, 1), C(4, -5, 4)
Solution
The position vectors bar"a", bar"b", bar"c" of the points A, B, C are
`bar"a" = 2hat"i" - hat"j" ,bar"b" = 4hat"i" + hat"j" + hat"k", bar"c" = 4hat"i" - 5hat"j" + 4hat"k"`
`bar"AB" = bar"b" - bar"a" = (4hat"i" + hat"j" + hat"k") - (2hat"i" - hat"j") = 2hat"i" + 2hat"j" + hat"k"`
`bar"BC" = bar"c" - bar"b" = (4hat"i" - 5hat"j" + 4hat"k") - (4hat"i" + hat"j" + hat"k") = - 6hat"j" + 3hat"k"`
`bar"CA" = bar"a" - bar"c" = (2hat"i" - hat"j") - (4hat"i" - 5hat"j" + 4hat"k") = - 2hat"i" + 4hat"j" - 4hat"k"`
∴ l(AB) = `|bar"AB"| = sqrt(2^2 + 2^2 + 1^2) = sqrt(4 + 4 + 1) = sqrt9 = 3`
l(BC) = `|bar"BC"| = sqrt((- 6)^2 + 3^2) = sqrt(36 + 9) = sqrt45 = 3sqrt5`
l(CA) = `|bar"CA"| = sqrt((- 2)^2 + 4^2 + (- 4)^2) = sqrt(4 + 16 + 16) = sqrt36 = 6`
∴ [l(AB)]2 + [l(CA)]2 = 32 + 62 = 9 + 36 = 45 = `(3sqrt5)^2`
= [l(BC)]2
∴ Δ ABC is right-angled at A.
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