Solution:

According to the question, the points are (2, 3), (5, 7), and (-3, -1). The area of triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is as follows:

= ½ [x₁(y₂ – y₃) + x₂(y₃ -y₁) + x₃(y₁ – y₂)]

The area of given triangle is as follows:

$ =\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }\left[ 2\left( 7+1 \right)\text{ }+\text{ }5\left( -1-3 \right)-3\left( 3-7 \right) \right] $

$ =~{\scriptscriptstyle 1\!/\!{ }_2}\text{ }\left[ 16-20\text{ }+\text{ }12 \right] $

$ =~{\scriptscriptstyle 1\!/\!{ }_2}\text{ }\left[ 8 \right] $

$ =\text{ }4 $

The triangle’s new coordinates are (3, 0), (6, 4), and (-2, -4) acquired by subtracting a point from the origin (-1, 3).

The new area of triangle is as follows:

$ =\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }\left[ 3\left( 4-\left( -4 \right) \right)\text{ }+\text{ }6\left( -4-0 \right)-2\left( 0-4 \right) \right] $

$ =~{\scriptscriptstyle 1\!/\!{ }_2}\text{ }\left[ 24-24+8 \right] $

$ =~{\scriptscriptstyle 1\!/\!{ }_2}\text{ }\left[ 8 \right] $

$ =\text{ }4 $

Because the area of the triangle before and after translation following the shifting of the origin is the same, i.e. 4.

The area of a triangle is said to be invariant to origin shifting.