Question

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

Answer 1

Hlo friend !!!

It is known that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Medians divide the triangles into two parts each of which is a triangle similar to the corresponding part of the other triangle, i.e. we have shown that the ratio of the length of the medians is equal to the ratio of one pair of sides of the original similar triangles. Hence, the areas of the two triangles are in the same proportion as the square of their medians.



P and Q are mid-points of BC and GF respectively.
Triangles ABC and EGF are similar. so ar(ABC)/ar(EGF)=(AB/EG)2ar(ABC)/ar(EGF)=(AB/EG)2  and triangles ABP and EGQ are similar, so AP/EQ=AB/EGAP/EQ=AB/EG which means ar(ABC)/ar(EGB)=(AP/EQ)2ar(ABC)/ar(EGB)=(AP/EQ)2
QED!

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Answer 2

Triangles. Prove that the ratio of the areas of two similar triangle is equal to the square of the ratio of their corresponding medians. Given: ∆ABC and ∆DEF such that ∆ABC ~ DEF. AP and DQ are medians drawn on sides BC and EF respectively.