. asked in Science & MathematicsMathematics · 1 decade ago

How do you use determinants to find the area of a triangle?

If someone could give me the formula?

or step by step

well the given vertices are A(1,1), B(6,6), C(2,10)

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  • Anonymous
    1 decade ago
    Favorite Answer

    Given: A triangle with vertices (1, 1), (6, 6), and (2, 10).

    Find the area of the triangle using determinants.

    The area (a) of a triangle = (1/2) base (b) times height (h)

    a = (±1/2) b h

    In determinants the answer can be negative or positive and area must be a positive therefore use ± if needed.

    Form a matrix with the x-values being the first column, the corresponding y-values being the second column, and the third column has all 1's

    Points: (1, 1), (6, 6), and (2, 10)

    area = ±1/2 times the matrix determinants

    | 1 1 1 |

    | 6 6 1 | (±1/2)

    |2 10 1|

    area = (±1/2)(uphill diagonals - downhill diagonals)

    area = (±1/2) times [ see next line ]

    [(1)(6)(1)+(1)(1)(2)+(1)(6)(10) -(1)(6)(2) -(1)(10)(1) -(1)(1)(6)]

    area = (±1/2)(6 + 2 + 60 - 12 - 10 - 6)

    area = (±1/2)(68 - 28)

    area = (±1/2)(40)

    area = 20 square units ANSWER

    --------------------------------------...

    CHECK

    After much computations of right triangles; I was able to create the triangle side lengths and calculate the area manually.

    Sides of triangel are 7.07, 9.06, and 5.66

    The height of the triangle is 4.42 if 9.06 is chosen as the base.

    area = 1/2 (base) (height)

    area = 1/2(9.06)(4.42)

    area = 20.02

    and it checks out.

    --------------------------------------...

    NOTE: The diagonals are written as follows

    For the matrix

    | a b c|

    | d e f |

    | g h i |

    Downhill Diagonals - Uphill Diagonals =

    aei + bfg + cdh - ceg - fha - ibd

    GOOD LUCK

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  • 3 years ago

    enable's shift A(-5 , a million) to the foundation or A(0 , 0) , the the rest vertices then improve into: B(5 , a million) , C(3 , x -a million) : then: If vertex A is placed on the foundation (0, 0) of a Cartesian coordinate equipment and the coordinates of the different 2 vertices are given via B = (x1, y1) and C = (x2, y2), then the section S could be computed as a million/2 cases truthfully the fee of the determinant: S = ½ |x1y2 ? x2y1| ½ |(5x - 5) ? 3| = 4 5x - 8 = 8 or 5x - 8 = -8 x = sixteen/5 or x = 0

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  • 1 decade ago

    you can use Hero's formula.

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