Question

prove that ratio of areas of two similar triangles equal to square of ratio of their corresponding medians

Answer 1

Let take triangle ABC similar to the triangle POR.
Here AD is median of triangle ABC.
So, BC=CD=1/2BC
Similarly take in triangle PQR
PS is the median of the triangle.
So, QS=RS=1/2QR

To.Proof.= Ratio of the two triangles equal to the ratio of the square of ratio of their corresponding medians

Proof= triangle ABC ~ Triangle PQR       (given)
<B=<Q   (corresponding angles of ~ triangles are equal) .......(1)
Also,
AB/PQ=BC/QR  (corresponding sides are ~ triangles are in same proportion)
AB/PQ=2BD/2QS
AB/PQ=BD/QS  (AS AD and PS are median) ........(2)
IN triangle ABD and triangle PQS
<B=<Q   From(1)
AB/PQ=BD/QS From (2)
Tri. ABD ~ Tri. PQS  (SAS)
Hence,
AB/PQ = AD/PS  (if tri. are ~ then there corresponding sides are also ~ and in proportion)  ...............(3)
Now,
since tri ABC ~ tri. PQR
we know that if two triangles are to be ~ then their crores. sides are to be the ratio of the square of their crores. sides
ar (ABC)/ar (PQR)=(AB)2/(PQ)2=(AB/PQ)2
then, ar (ABC)/ar (PQR)=(AD)2/(PS)2=(AD/PS)2

Hence proved.

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