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Question
Find the area of the triangle ABC with the coordinates of A as (1, −4) and the coordinates of the mid-points of sides AB and AC respectively are (2, −1) and (0, −1).
Solution
Let D is mid-point of AB and E is the midpoint of AC.
Coordinates of A (1, -4)
Using mid-point `"x" = ("x"_1+"x"_2)/2 , "y" = ("y"_1+"y"_2)/2`
we will find coordinates of B using mid-point in AB
`2 = (1+"x"_2)/2 , -1 = (-4+"y"_2)/2`
`"x"_2 = 3, "y"_2 = 2`
Coordinates of B (3,2)
Now, we will find coordinates of C from mid-point formula in AC
`0 = (1+"x"_3)/2, -1 =(-4+"y"_3)/2`
`"x"_3 = -1, "y"_3 = 2`
Coordinates of C (-1,2)
Now, using coordinates A (1,-4), B(3,2) and C(-1,2) in the formula of area of triangle which is:
`1/2["x"_1("y"_2-"y"_3)+"x"_2("y"_3-"y"_1)+("x"_3("y"_1-"y"_2)]`
`1/2 [1(2-2)+3(2-(-4))+(-1)(-4-2)]`
`=1/2 [0+18+6]`
`1/2(24) = 12 "sq.units"`
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