If areas of two similar triangles are equal.prove that they are congru

1.	If areas of two similar triangles are equal.prove that they are congruent.
Answer» Given :\xa0{tex}\\triangle \\mathrm{ABC}{\\sim} \\triangle \\mathrm{PQR}{/tex}\xa0&ar\xa0{tex}\\triangle \\mathrm{ABC}{/tex}\xa0= ar\xa0{tex}\\Delta \\mathrm{PQR}{/tex}{tex}{/tex}To prove:\xa0{tex}\\triangle \\mathrm{ABC} \\cong \\triangle \\mathrm{PQR}{/tex}Since,\xa0{tex}\\triangle \\mathrm{ABC}{\\sim} \\Delta \\mathrm{PQR}{/tex}ar\xa0{tex}\\triangle A B C=\\text { ar } \\triangle P Q R{/tex}\xa0(given){tex}\\frac{{\\Delta ABC}}{{ar\\Delta PQR}} = 1{/tex}{tex}\\Rightarrow{/tex}\xa0{tex}\\frac{{A{B^2}}}{{P{Q^2}}} = \\frac{{B{C^2}}}{{Q{R^2}}} = \\frac{{C{A^2}}}{{P{R^2}}} = 1{/tex}[Using Theorem of area of similar Triangles]{tex}\\Rightarrow{/tex}\xa0AB = PQ, BC = QR & CA = PRThus,\xa0{tex}\\triangle \\mathrm{ABC} \\cong \\triangle \\mathrm{PQR}{/tex}

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