Given that Δ ABC ∼ Δ PQR, CM and RN are respectively the medians of Δ ABC and Δ PQR Prove that

i. Δ AMC ∼ Δ PNR

ii.

iii. ΔCMB ∼ ΔRNQ

(i) Given, Δ ABC ∼ Δ PQR

So, ………eq(1)

And ∠ A = ∠ P, ∠ B = ∠ Q and ∠ C = ∠ R ……..eq(2)

⇒ As CM and RN are medians

⇒ AB = 2AM and PQ = 2PN

From eq(1) we have

⇒

i.e., ……..eq(3)

Also, from eq(2) Δ MAC = Δ NPR ……eq(4)

⇒ From eq(3) and eq(4) we have

⇒ Δ AMC ∼ Δ PNR …..eq(5)

⇒ By SAS similarity if one angle of a triangle is equal to another angle of a triangle and the including sides of the these angles are proportional, then the two triangles are similar.

(ii) From eq(5) we have ……..eq(6)

⇒ From eq(1) we have,

⇒ …….eq(7)

⇒ From eq(6) and eq(7) we have

⇒ …..eq(8)

(iii) Again from eq(1) we have

⇒

⇒ From eq(8) we have

⇒

⇒

i.e., …..eq(10)

⇒ From eq(9) and eq(10) we have

⇒

∴ Δ CMB ∼ Δ RNQ