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Problem Which of the following expressions gives the area of the triangle with vertices ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) (x_1,y_1),\,(x_2,y_2),\,(x_3,y_3) ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) 1 2 ( x 1 ( y 2 − y 3 ) + x 2 ( y 3 − y 1 ) + x 3 ( y 1 − y 2 ) ) \dfrac12\big(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\big) 2 1 ( x 1 ( y 2 − y 3 ) + x 2 ( y 3 − y 1 ) + x 3 ( y 1 − y 2 ) ) 1 4 ∣ ∣ x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 ∣ ∣ \dfrac14\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right| 4 1 x 1 x 2 x 3 y 1 y 2 y 3 1 1 1 1 2 ∣ ∣ x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 ∣ ∣ \dfrac12\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right| 2 1 x 1 x 2 x 3 y 1 y 2 y 3 1 1 1 ∣ ∣ x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 ∣ ∣ \left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right| x 1 x 2 x 3 y 1 y 2 y 3 1 1 1 Browse the problem library