Question

Prove that four triangles obtained by joining the mid-point of three sides of A triangle are similar to the original triangle

Answer 1

That is Triangle in the picture... Its not cropped in the correct way am sorry about it, but this is the answer!

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

Hope you understand