Plot each point of the following points, and connect them to form the triangle ABC. 

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).   Prove that both methods yield the same results

  • A(4,-3);  B(4, 1);  C(2,1)

Expert Answers

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You should use the following formula to evaluate the area of triangle such that:

`S = (1/2)|[(x_A,y_A,1),(x_B,y_B,1),(x_C,y_C,1)]|`

You should substitute the values of coordinates of vertices such that:

`S = (1/2)|[(4,-3,1),(4,1,1),(2,1,1)]|` = `(1/2)|(4 + 4 - 6 - 2 - 4 + 12)|`

`S = 8/2 => S = 4`

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You should use the following formula to evaluate the area of triangle such that:

`S = (1/2)|[(x_A,y_A,1),(x_B,y_B,1),(x_C,y_C,1)]|`

You should substitute the values of coordinates of vertices such that:

`S = (1/2)|[(4,-3,1),(4,1,1),(2,1,1)]|` = `(1/2)|(4 + 4 - 6 - 2 - 4 + 12)|`

`S = 8/2 => S = 4`

You may also use Heron's formula to evaluate the area of triangle but you should find the lengths of sides of triangle such that:

`a = sqrt((x_C - x_B)^2+(y_C - y_B)^2)`

`a = sqrt((2-4)^2 + (1-1)^2) => a = 2`

`b = sqrt((x_C - x_A)^2+(y_C - y_A)^2)`

`b = sqrt((2-4)^2+(1+3)^2) => b = sqrt20 = 2sqrt5`

`c = sqrt((x_B - x_A)^2 + (y_B-y_A)^2)`

`c = sqrt((4-4)^2 + (1+3)^2) => c = 4`

Hence, using the Heron's formula yields:

`S = (1/4)sqrt((a+b+c)(a+b-c)(a+c-b)(b+c-a))`

`S= (1/4)sqrt((6+2sqrt5)(2sqrt5 - 2)(6-2sqrt5)(2sqrt5+ 2))`

`S= (1/4)sqrt((6^2 - (2sqrt5)^2)((2sqrt5)^2 - 2^2))`

`S = (1/4)sqrt((36 - 20)(20 - 4)) => S = (1/4)sqrt(16*16)`

`S = 16/4 => S = 4`

Hence, evaluating the area of triangle, under the given conditions, yields `S = 4` .

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