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

Let P(4,6), Q(7,10) and R(-1,2) be the given points.

∴ Area of ∆PQR = 1/2 [4(10 - 2) + 7(2 - 6) - (-1)(6 - 10)]

=1/2 [32 - 28 + 4] = 4 sq. units

Now shifting (x, y) to (X - 2, Y + 1)

New coordinates are X = x + 2, Y = y - 1

∴ P(4,6) → (6,5)

   Q(7,10) → (9,9)

    R(-1,2) → (1,1)

∴  Area of ∆ = 1/2 [6(9 - 1) + 9(1 - 5) + 1(5 - 9)]

   = 1/2 [48 - 36 - 4] = 4 sq. units

Hence, the area remains invariant.