For three points A = (x1, y1), B = (x2, y2), C = (x3, y3) in...

For three points A = (x1, y1), B = (x2, y2), C = (x3, y3) in the plane, the area of triangle ABC is equal to: $frac{1}{2} det egin{bmatrix} x_1 & y_1\ x_2 & y_2\ x_3 & y_3 end{bmatrix}$ $det egin{bmatrix} x_2 - x_1 & x_3 - x_1\ y_2 - y_1 & y_3 - y_1 end{bmatrix}$ $frac{1}{2} left| det egin{bmatrix} 1 & x_1 & y_1\ 1 & x_2 & y_2\ 1 & x_3 & y_3 end{bmatrix} ight|$

Transcript

00:01 We are given the three points in the plane, that is a is x1y1, b is x2y2 and c is x3y3 and we determine the area of the triangle a, b ,c.

00:14 So using determinants, we can write the area of the triangle a, b, c, and that equals half times the absolute value of the determinant, x1, y1 and 1 in the first row.

00:41 Then x2 y2 and 1 in the second row then x3 y 3 and 1 in the third row so we should find the determinant of this one and then we take the absolute value and multiply by half so this is the formula to determine the area of the triangle using determinants but when we look at this answer choices we do not see any of the answer choices matching with this one.

01:10 However, the last one seems a little bit closer, but the columns are different.

01:16 So, i'm going to do some re -rating of this determinant using the properties of the determinant.

01:24 That is, if we interchange any two rows or any two columns of a determinant, the value of the determinant gets multiplied by negative one.

01:34 So we are going to use that property so that we can rewrite this a determinant so here i put half and then we have absolute value now we still have to find the determinant i'm going to interchange these two columns so when we interchange these two columns the determinant gets multiplied by negative so i'm going to put a negative here and then write this after interchanging the column two and column three then the column one remains as it is so i write the column 1 as it is and then if you interchange the column 2 and column 3 the column 3 becomes column 2 elements so here i have to put 1 and then the column 2 elements will become column 3 elements so here i write this is y1 then y2 and y 3 so we have to find this determine and the absolute value now once again we are going to interchange the first row with the first column with the second column...

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