Question

Verify that the area of the triangle with vertices (2.
3). (5, 7) and (-3,-1) remains invariant under the
translation of the axes when the origin is shifted to
the point (-1,3)​

Answer 1

Answer:

The are of the triangle remains invariant. (Proved)

Step-by-step explanation:

The vertices of the triangle are A(2,3), B(5,7), and C(-3,-1).

So, the area of the triangle ΔABC will be,  

$\frac{1}{2}[2(7+1)+5(-1-3)-3(3-7)]$

=$\frac{1}{2}[16-20+12]$

=$\frac{1}{2} [8]$

=4 square units

Now, we are shifting the origin to the point (-1,3), and the new coordinate axes  X' and Y' are remains parallel to X and Y axes respectively.

So, in the new coordinate plane, the X values of the coordinates of points A, B, C will be added by 1 and the Y values of the coordinates of points A, B, C will be deducted by 3.

Hence, the new coordinates of A, B, C will be (2+1,3-3) ≡ (3,0), (5+1,7-3) ≡ (6,4)and (-3+1,-1-3) ≡ (-2,-4) respectively.

So, the area of triangle ΔABC with the new coordinates of A, B, C will be

$\frac{1}{2}[3(8)+6(-4-0)-2(0-4)]$

=$\frac{1}{2}[24-24+8]$

=4 square units.

Therefore, the are of the triangle remains invariant. (Proved)