prove that the area of triangle is invariant under the transformation of the axes .

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$$\begin{gathered} \mathrm{When} \mathrm{you} \mathrm{transform} \mathrm{axes}, \mathrm{you} \mathrm{can} \mathrm{say} \mathrm{that} \mathrm{line} \mathrm{associated} \mathrm{with} \mathrm{the} \mathrm{old} \mathrm{axis} \mathrm{gets} \mathrm{translated} \mathrm{to} \\ \mathrm{a} \mathrm{new} \mathrm{position}. \\ \mathrm{And} \mathrm{as} \mathrm{we} \mathrm{know} \mathrm{that} \mathrm{under} \mathrm{a} \mathrm{translation}, \mathrm{distance} \mathrm{between} \mathrm{two} \mathrm{points} \mathrm{remains} \mathrm{the} \mathrm{same}, \mathrm{only} \\ \mathrm{the} \mathrm{position} \mathrm{of} \mathrm{points} \mathrm{is} \mathrm{changed}. \mathrm{Thus}, \mathrm{the} \mathrm{line} \mathrm{segments} \mathrm{are} \mathrm{mapped} \mathrm{to} \mathrm{line} \mathrm{segments} \mathrm{and} \mathrm{a} \\ \mathrm{triangle} \mathrm{remains} \mathrm{the} \mathrm{same} \mathrm{triangle} \mathrm{but} \mathrm{shifts} \mathrm{its} \mathrm{position} \mathrm{and} \mathrm{hence} \mathrm{the} \mathrm{area} \mathrm{of} \\ \mathrm{the} \mathrm{triangle} \mathrm{remains} \mathrm{same} \mathrm{after} \mathrm{transformation} \mathrm{of} \mathrm{axes}. \end{gathered}$$

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