Given points A(2m-1,1), B(1-2m,3), C(m-1,m+2) what is m if area of triangle ABC is the  largest area?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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The problem provides the coordinates of vertices of triangle, hence, you need to evaluate the area of triangle using the following determinant formula, such that:

`A_(ABC) = (1/2)|[(2m-1,1,1),(1-2m,3,1),(m-1,m+2,1)]|`

`A_(ABC) = (1/2)|3(2m-1) + (1-2m)(m+2) + m - 1 - 3(m - 1) - (m+2)(2m-1) - 1 + 2m|`

`A_(ABC) = (1/2)|6m - 3 + m + 2 - 2m^2 - 4m + m - 1 - 3m + 3 - 2m^2 + m - 4m + 2 - 1 + 2m|`

`A_(ABC) = (1/2)|-4m^2 - 6m + 2|`

`A_(ABC) = 2m^2 + 3m + 1`

The area of triangle ABC is maximum if `A'(m) = 0` , such that:

`A'(m) = 4m + 3 => 4m + 3 = 0 => m = -3/4`

`A(-3/4) = |2*9/16 - 9/4 + 1|`

`A(-3/4) = |9/8 - 9/4 + 1| => A(-3/4) = |(9 - 18 + 8)/8| => A(-3/4) = 1/8`

Hence, evaluating the value of m, under the given conditions, yields `m = -3/4.`

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