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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).

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Solution

Let the vertices of a triangle be A(2,3),B(5,7),and ~C(-3,-1)Then, area of ΔABC is given byΔ=12|x1(y2y3)+x2(y3y1)+x3(y1y2)|=12|2(7+1)+5(13)3(37)|12|2×8+5×(4)3×(4)|12|1620+12|=82=4Δ=4 sq.unit.It is given that the origin is shifted at (-1,3).Then neq coordinates of the vertices are A1=(21,3+3)=(+1,6)B1=(41,7+3)=(4,10)and C1=(31,1+3)=(4,2)Therefore the area of the triangle in the neq coordinate system is given byΔ=12|[1(102)+4(26)4(610)]|=12|[1×8+4×(4)4×(4)]|=12|816+16|=12|8|=82Δ1=4 sq.units ...(2)From(i) and (ii),we getΔ=Δ1Hence,the area of a triangle is invariant under the translation of the axes.


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