Find the area of a triangle with vertices A(1,3,5), B(-2,-3,-4) and c(0,3,-1).

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We have to find the area of the triangle with vertices A(1, 3, 5) , B(-2, -3 , -4) and C(0, 3 , -1).

The length of the sides are:

AB = sqrt [( 1 + 2)^2 + ( 3 + 3)^2 + (5 + 4)^2]

=> sqrt ( 9 + 36 + 81)

=> sqrt 126

BC = sqrt [( 0 + 2)^2 + ( 3 + 3)^2 + (-1 + 4)^2]

=> sqrt ( 4 + 36 + 9)

=> sqrt 49

=> 7

CA = sqrt [( 1 - 0)^2 + (3 - 3)^2 + ( 5 + 1)^2]

=> sqrt ( 1 + 36)

=> sqrt 37

The area of the triangle can be found by using Heron's formula:

A = sqrt [s( s - a)(s - b)(s - c)], where a, b and c are the length of the sides and s is the semi-perimeter.

s = (7 + sqrt 37 + sqrt 126)/2

A = sqrt [((7 + sqrt 37 + sqrt 126)/2)((7 + sqrt 37 + sqrt 126)/2 - sqrt 126)((7 + sqrt 37 + sqrt 126)/2 - sqrt 37)((7 + sqrt 37 + sqrt 126)/2 - 7)]

=> A = sqrt ( 353.25)

=> A = 18.794

The required area of the triangle is 18.794

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