Question

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

(i) Two circles with different radii are similar. 

(ii) any two rectangles are similar 

(iii) if two triangles are similar then their corresponding angles are equal and their corresponding sides are equal 

(iv) The length of the line segment joining the midpoints of any two sides of a triangles is equal to half the length of the third side. 

(v) In a △ABC , AB = 6 cm, ∠A = 45° and AC = 8 cm and in a △DEF , DF = 9 cm ∠D = 45° and DE = 12 cm then △ABC ~ △DEF.

Answer 1

(i) Two rectangles are similar if their corresponding sides are proportional. 

(ii) True Two circles of any radii are similar to each other. 

(iii) false If two triangles are similar, their corresponding angles are equal and their corresponding sides are proportional. 

(iv) True Suppose ABC is a triangle and M, N are

Construction: DE is expanded to F such that EF = DE 

To proof = DE = 1/2 BC 

Proof: In ∆ADE and ∆CEF 

AE = EC (E is the mid point of AC) 

DE = EF (By construction) 

AED = CEF (Vertically Opposite angle) 

By SAS criterion, ∆ADE ~ ∆CEF 

CF = AD (CPCT) 

⇒ BD = CF 

∠ADE = ∠EFC (CPCT) 

Since, ∠ADE and ∠EFC are alternative angles 

Hence, AD ‖ CF and BD ‖ CF 

When two sides of a quadrilateral are parallel, then it is a parallelogram 

∴ DF = BC and BD ‖ CF 

∴BDFC is a parallelogram 

Hence, DF = BC 

⇒ DE + EF = BC 

⇒ DE = 1/2 BC 

(v) False In ∆ABC, AB = 6 cm, ∠A = 45° and AC = 8 cm and in ∆DEF,DF = 9 cm, ∠D = 45° and DE = 12 cm, then ∆ABC ~ ∆DEF. 

In ∆ABC and ∆DEF