Verify that the area of the triangle with vertices (4,6), (7,10) and (1,-2) remains invariant under the translation of axes when the origin is shifted to the point (-2,1).

Explanation

The area of a triangle is invariant to shifting of origin. The area of triangle with vertices (x_1, y_1), (x_2, y_2), and (x_3, y_3) is `= 1/2 [x_1(y_2 – y_3) + x_2(y_3 -y_1) + x_3(y_1 – y_2)]`

Hence, the area of given triangle `= 1/2[4(10-(-2)) + 7(-2-6) + 1(6-10)] = 1/2 [48 – 56 - 4] => -6`

we takes modulus value of `-6` i.e. `6`.

The area of the triangle before and after the translation after shifting of origin remains same, i.e. `6` sq. units, therefore we can say that the area of a triangle is invariant to shifting of origin.