Question

Triangle is a very popular shape used in interior designing. The picture given above shows a cabinet designed by a famous interior designer. Here the largest triangle is represented by △ ABC and smallest one with shelf is represented by △ DEF. PQ is parallel to EF.

(i) Show that △ DPQ ∼ △ DEF.

(ii) If DP= 50 cm and PE = 70 cm then find PQ/EF.

(iii) (A) If 2AB = 5DE and △ ABC ∼ △ DEF then show that perimeter of △ABC/perimeter of △DEF is constant.

OR

(iii) (B) If AM and DN are medians of triangles ABC and DEF respectively then prove that △ ABM ∼ △ DEN.

Answer 1

(i) ∠DPQ = ∠DEF 

∠PDQ =∠EDF 

 Therefore DPQ ∼ DEF 

(ii) DE = 50 + 70 = 120 cm 

DP/DE = PQ/EF

Therefore PQ/EF= 50/120 or 5/12

 (iii) (A)

AB/DE = 5/2 = BC/EF = AC/DF 

⇒ AB = 5/2 DE 

perimeter of △ABC/perimeter of △DEF = 5/2 (DE + EF + FD)/ (DE + EF+ FD) = 5/2 (Constant)

 (iii) (B)  

AB/DE = BC/EF= (BC/2)/(EF/2) = BM/EN 

Also ∠B =∠E 

Therefore △ ABM ∼ △ DEN.