Question

सिद्ध कीजिए कि दो समरूप त्रिभुजों के क्षेत्रफलों का अनुपात इनकी संगत माध्यिकाओं के अनुपात का वर्ग होता है।

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Solution

∵

increment ABC tilde increment DEF

rightwards double arrow

fraction numerator ar left parenthesis increment ABC right parenthesis over denominator ar left parenthesis increment DEF right parenthesis end fraction equals AB squared over DE squared

...(i)

AB over DE equals BC over EF space space space space space space space space space left square bracket because space increment ABC space tilde space increment DEF right square bracket

rightwards double arrow

AB over DE equals fraction numerator 2 BP over denominator 2 EQ end fraction

rightwards double arrow

AB over DE equals BP over EQ

∆ABP और ∆DEQ

AB over DE equals BP over EQ

और

angle straight B space equals space angle straight E space space space space space space space space left square bracket because space increment ABC space tilde space increment DEF right square bracket

Therefore, by using SAS similar condition

increment ABP tilde increment DEQ

rightwards double arrow space space space space AB over DE equals AP over DQ rightwards double arrow space space space space AB squared over DE squared equals AP squared over DQ squared space space... left parenthesis ii right parenthesis

समीकरण (i) और (ii) की तुलना करने पर

fraction numerator ar left parenthesis increment ABC right parenthesis over denominator ar left parenthesis increment DEF right parenthesis end fraction equals AP squared over DQ squared

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