Question

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

Answer 1

Step-by-step explanation:

Given:- Let △ABC∼△PQR

AD and PS are corresponding medians.

To prove:- ar(△PQR)ar(△ABC)=(PSAD)2

Proof:- In △ABC 

∵AD is median.

∴BD=CD=21BC

Similarly, in △PQR

PS is median.

∴QS=RS=21QR

Now,

△ABC∼△PQR(Given)

∠B=∠Q.....(1)(Corresponding angles of similar triangles are equal)

PQAB=QRBC(Corresponding sides of similar triangles are proportional)

⇒PQAB=2QS2BD(∵AD and PS are medians)

PQAB=QSBD.....(2)

Now, in △ABD and △PQS,

∠B=∠Q(From (1))

PQAB=QS