Question

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

Answer 1

$\text{Let the vertices of a triangle be A(2,3),B(5,7),and ~C(-3,-1)}Then,areaof\Delta ABCisgivenby\Delta =\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|=\frac{1}{2}|2(7+1)+5(-1-3)-3(3-7)|\frac{1}{2}|2\times 8+5\times (-4)-3\times (-4)|\frac{1}{2}|16-20+12|=\frac{8}{2}=4\Rightarrow \Delta =4sq.unit.\text{It is given that the origin is shifted at (-1,3).Then neq coordinates of the vertices are}{A}_1=(2-1,3+3)=(+1,6)B_1=(4-1,7+3)=(4,10)and{C}_1=(-3-1,-1+3)=(-4,2)\text{Therefore the area of the triangle in the neq coordinate system is given by}\Delta =\frac{1}{2}|\left[1\right(10-2)+4(2-6)-4(6-10\left)\right]|=\frac{1}{2}|[1\times 8+4\times (-4)-4\times (-4\left)\right]|=\frac{1}{2}|8-16+16|=\frac{1}{2}|8|=\frac{8}{2}\Rightarrow {\Delta }_1=4sq.units...\left(2\right)From\left(i\right)and\left(ii\right),weget\Delta ={\Delta }_1\text{Hence,the area of a triangle is invariant under the translation of the axes.}$