Plot each point  using two techniques Heron's Formula, the three-point coordinate formula A(-2,5); B(1, 3); C(-1,0)Plot each point of the following points, and connect them to form the triangle...

Plot each point  using two techniques Heron's Formula, the three-point coordinate formula A(-2,5); B(1, 3); C(-1,0)

Plot each point of the following points, and connect them to form the triangle ABC.  Find the area of the triangle using two techniques;  Heron's Formula, and the three-point coordinate formula (see Math Open Reference, 2009).  A(-2,5);  B(1, 3);  C(-1,0)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should evaluate the area of triangle using the determinant formed by the coordinates of vertices of triangle such that:

`S = (1/2) |[(-2,5,1),(1,3,1),(-1,0,1)]|`

`S = (1/2)(-6 - 5 + 3 - 5) => S = |-13|/2 => S = 6.5`

You may evaluate the area of triangle using the alternative formula called Heron's formula such that:

`S = sqrt((a+b+c)/2(a+b-c)/2(a+c-b)/2(b+c-a)/2)`

Notice that a,b,c represent the lengths of sides of triangle, hence, you need to use distance formula to evaluate the lengths such that:

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

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

`[AB] = sqrt(9 + 4) => [AB] = sqrt 13`

`[AC] = sqrt((-1+2)^2 + (0-5)^2) => [AC] = sqrt(1 + 25) => [AC] = sqrt 26`

`[BC] = sqrt((-1-1)^2 + (0-3)^2) => [BC] = sqrt(4 + 9) => [BC] = sqrt 13`

`S = (1/4)sqrt((2sqrt13 + sqrt26)(2sqrt13-sqrt26)(sqrt13+sqrt26-sqrt13)(sqrt13+sqrt26-sqrt13))`

`S = (26/4) => S = 13/2 => S = 6.5`

Hence, evaluating the area using the two formulas yields `S = 6.5.`

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