Question

For each of the following statements state whether true(T) or false (F) 

the ratio of the perimeter of two similar triangles is the same as the ratio of the their corresponding medians.

Answer 1

True 

Given: ΔABC ~ ΔDEF 

To prove=`(Ar(ΔABC))/(Ar(ΔDEF))=((AP)/(DQ))^2` 

Proof: in ΔABP and ΔDEQ
∠𝐵𝐴𝑃= ∠𝐸𝐷𝑄 (𝐴𝑠 ∠𝐴= ∠𝐷,𝑠𝑜 𝑡ℎ𝑒𝑖𝑟 𝐻𝑎𝑙𝑓 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑒𝑞𝑢𝑎𝑙)
∠𝐵= ∠𝐸 (∠𝐴𝐵𝐶 ~ Δ𝐷𝐸𝐹)
By AA criterion, ΔABP and ΔDEQ 

`(AB)/(DE)=(AP)/(DQ)`        .................(1) 

Since, ΔABC ~ ΔDEF 

∴ `(Ar(ΔABC))/(Ar(ΔDEF))=((AB)/(DE))^2` 

⇒ `(Ar(ΔABC))/(Ar(ΔDEF))=((AP)/(DQ))^2`           [𝑈𝑠𝑖𝑛𝑔 (1)]