• If the area of a triangle is given, then both positive and negative values of the determinant are used for calculation.why?

Determinant can have both positive and negative values, but area is always positive.
Consider a simple example:

Let us find the coordinate of  a missing point of a triangle, when the area is known.
Let A(-2,2);B(1,5) and C(x,-1) be the three coordinates and area of the triangle =16.5
Now,Area =±12-221151x-1116.5=±12[-2(5+1)-1(2+1)+x(2-5)]33=±[-15-3x]11=±[-5-x]Now, -5-x=11  or 5+x=11x=-16     or x=6Now, x=-16 is rejected as it is giving wrong areaSo, we obtain  x=6 ,by taking negative sign

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