Question

9. Prove that the three line segments which join
the mid-points of the sides of triangle, divide
it into four triangles which are congruent to
each other.
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Answer 1

Step-by-step explanation:

To prove : △ABC ~ △DEF 

△ABC ~ △ADF

△ABC ~ △BDE

△ABC ~ △EFC

Proof: In △ABC, D and E are mid points AB and AC resoopectively.

∴ DF | | BC(midpoint theorem) 

In △ABC = △ADF  

∠A is common;  ∠ADF = ∠ABC (corresponding angles)

△ABC ~ △DF (AA similarity) .......(1) 

Similarly we can prove △ABC ~ △BDE (AA similarity) .......(2)

△ABC ~ △EFC (AA similarity)........(3) 

In △ABC and △DEF;

since D,E,F are the midpoints AB, BC and AC respectively.

DF = (1/2) × BC; DE = (1/2) × AC; EF = (1/2) ×AB; (midpoint theorem)    

∴ AB = BC = CA = 2

EF = DF = DE

∴ △ABC ~ △EFD (SSS similarity)...........(4)

From(1),(2),(3) and (4)

△ABC ~ △DEF

△ABC ~ △ADF

△ABC ~ △BDF

△ABC ~ △EFC