Question

For the following statement, state whether true (T) or false (F): 

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

Answer 1

This statement is true.

Explanation:

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)]