D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.

Name: D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.
Uploaded: Fri, 2021-01-08 02:02
Description: D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.

D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.

Answers (1)

Solution.

Given: In $\triangle ABC$ , D, E and F are respectively the mid-points of the sides AB, BC and CA.
To prove: $\triangle ABC$ is divided into four congruent triangles.

Proof: Using given conditions we have
$AD=BD=\frac{1}{2}AB,BE=EC=\frac{1}{2}BC,AF=CF=\frac{1}{2}AC$
Using mid-point theorem
$EF\parallel AB$ and $EF=\frac{1}{2}AB=AD=BD$
$ED\parallel AC$ and $ED=\frac{1}{2}AC=AF=CF$
$DF\parallel BC$ and $DF=\frac{1}{2}BC=BE=EC$
In $\triangle ADF$ and $\triangle EFD$
AD = EF
AF = DE
DF = FD {common side}
$\triangle ADF\cong \triangle EFD$ {by SSS congruence}
Similarly we can prove that,
$\triangle DEF\cong \triangle EDB$
$\triangle DEF\cong \triangle CFE$
So, $\triangle ABC$ is divided into four congruent triangles
Hence proved