Mathematics

Mathematics

Problem

Which of the following expressions gives the area of the triangle with vertices (x1,y1), (x2,y2), (x3,y3)(x_1,y_1),\,(x_2,y_2),\,(x_3,y_3)? a) 12(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))\dfrac12\big(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\big) b) 14∣∣x1y11x2y21x3y31∣∣\dfrac14\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right| c) 12∣∣x1y11x2y21x3y31∣∣\dfrac12\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right| d) ∣∣x1y11x2y21x3y31∣∣\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right|

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