Question

Prove that the line segment joining the mid-points of the sides of a triangle form four
triangles, each of which is similar to the original triangle​

Answer 1

Answer:

here is your solution check one by one pl

Answer 2

△ABC ~ △DEF

△ABC ~ △ADF

△ABC ~ △BDF

△ABC ~ △EFC

Step-by-step explanation:

To prove : △ABC ~ △DEF

△ABC ~ △BDE

△ABC ~ △EFC

Proof:

In △ABC, D and E are mid points AB and AC respectively.

∴ 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 = $\frac{1}{2}\times AB$ ; (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

hence proved

#Learn more:

In a figure ,P and Q are two points on equal sides AB and AC an isosceles triangle ABC such that AP = AQ,rove that BQ = CF

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