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(y2−y3)+x2(y3−y1)+x3(y1−y2)|=12|2(7+1)+5(−1−3)−3(3−7)|12|2×8+5×(−4)−3×(−4)|12|16−20+12|=82=4⇒Δ=4 sq.unit.It is given that the origin is shifted at (-1,3).Then neq coordinates of the vertices are A1=(2−1,3+3)=(+1,6)B1=(4−1,7+3)=(4,10)and C1=(−3−1,−1+3)=(−4,2)Therefore the area of the triangle in the neq coordinate system is given byΔ=12|[1(10−2)+4(2−6)−4(6−10)]|=12|[1×8+4×(−4)−4×(−4)]|=12|8−16+16|=12|8|=82⇒Δ1=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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