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# Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians. Q.6

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What is the best way to solve the problem of exercise 6.4 of chapter triangles, i don’t know how to solve it so please help me for solving question Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.

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1. Given: AM and DN are the medians of triangles ABC and DEF respectively and Î”ABC ~ Î”DEF.

We have to prove: Area(Î”ABC)/Area(Î”DEF) = AM2/DN2

Since, Î”ABC ~ Î”DEF (Given)

âˆ´ Area(Î”ABC)/Area(Î”DEF) = (AB2/DE2) â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦(i)

and, AB/DE = BC/EF = CA/FD â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦(ii)

In Î”ABM and Î”DEN,

Since Î”ABC ~ Î”DEF

âˆ´ âˆ B = âˆ E

AB/DE = BM/EN [Already Proved in equationÂ (i)]

âˆ´ Î”ABC ~ Î”DEF [SAS similarity criterion]

â‡’ AB/DE = AM/DN â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(iii)

âˆ´ Î”ABM ~ Î”DEN

As the areas of two similar triangles are proportional to the squares of the corresponding sides.

âˆ´ area(Î”ABC)/area(Î”DEF) = AB2/DE2Â = AM2/DN2

Hence, proved.

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