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Finding the area of a triangle based on vertices lines?

I can’t solve this, could someone please help me?

16) The lines: x-2y=2, -3x+y=4, and 2x+y=4 intersect in pairs to determine the vertices of a triangle. Find the area of the triangle.

Draw all three lines on x-y plane, and find all intersection points: (0,4), (2,0) and (-2,-2).
Since the triangle can be split into two triangles with a common base = 2+4/3 = 10/3, the area is equal to
(1/2)(10/3)(4+2) = 10 units^2
———
Attn: Try to find to simpler way to get it done.

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2 Responses to “Finding the area of a triangle based on vertices lines?”

  1. Draw all three lines on x-y plane, and find all intersection points: (0,4), (2,0) and (-2,-2).
    Since the triangle can be split into two triangles with a common base = 2+4/3 = 10/3, the area is equal to
    (1/2)(10/3)(4+2) = 10 units^2
    ———
    Attn: Try to find to simpler way to get it done.
    References :

  2. -3x+y=4, and 2x+y=4 intersect at (0,4), call that point A
    x-2y=2 and 2x+y=4 intersect at (2,0), call that point B
    x-2y=2 and -3x+y=4 intersect at (-2,-2), call that point C
    Side AB is on line 2x+y=4, which has slope -2
    Side BC is on line x-2y=2, which has slope 1/2
    AB and BC are perpendictular (-2 * 1/2 = -1)
    The length of AB is sqrt[20]
    The length of BC is sqrt[20]
    The area of ABC is sqrt[20] * sqrt[20] / 2 = 10
    References :

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