Statement: Ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Given: Two triangles ABC and PQR such that △ABC∼△PQR
To prove: area(ABC)area(PQR)=(AB2PQ)=(BCPQ)2=(CARP)2
Proof: For finding the areas of the two triangles,we draw altitudes AM and PN of the triangles.
Now, area(ABC)=12BC×AM
And area(PQR)=12QR×PN
So, area(ABC)area(PQR)=12×BC×AM12×QR×PN=BC×AMQR×PN.....(1)
Now, in △ABC and △PQN
∠B=∠Q (As △ABC∼△PQR)
and ∠M=∠N (Each=90o)
so, △ABM∼△PQN (AA similarity criterion)
Therefore AMPN=ABPQ.....(2)
Also, △ABC∼△PQR
ABPQ=BCQR=CARP.......(3)
area(ABC)area(PQR)=ABPQ×AMPN (from (1) and (3))
Therefore =ABPQ×ABPQ=(ABPQ)2 (from (2))
Now using (3) we get
area(ABC)area(PQR)=(AB2PQ)=(BCPQ)2=(CARP)2
![666498_626122_ans_a0a428810c594d9587c4a05f60f82932.PNG](https://search-static.byjusweb.com/question-images/toppr_ext/questions/666498_626122_ans_a0a428810c594d9587c4a05f60f82932.PNG)