Question

if 2 triangles are equiangular prove that the ratio of the corresponding medians are in the same ratioas the ratio of the corresponding sides

Answer 1

Given: △ABC ~ △DEF. AP is the median to side BC of △ABC and DQ is the median to side EF of △DEF.

ACDF=BCEF {Corresponding sides of similar triangles are proportional}

⇒ACDF=2PC2QF=PCQF (1)

{P is the mid-point of BC and Q is the mid-point of EF}



To Prove: ar(△ABC)ar(△DEF)=AP2DQ2
Proof: ar(△ABC)ar(△DEF)=BC2EF2
{The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides}

⇒ar(△ABC)ar(△DEF)=(2PC)2(2QF)2=PC2QF2 (2)



In △APC and △DQF
ACDF=PCQF from (1)

And, ∠C=∠F {Corresponding angles of similar triangles are equal}



Therefore, by SAS similarity criterion, △APC ~ △DQF
Therefore, APDQ=PCQF (3)



Putting (3) in (2), we get

ar(△ABC)ar(△DEF)=AP2DQ2


Hence Proved