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Question

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

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Solution

## Given in a ΔABC,D,E and F respectively the mid-points of the sides AB, BC and CA. To prove ΔABC is divided into four congruent triangles. Proof Since, ABC is a triangle and D, E and F are the mid-points of sides AB, BC and CA, respectively Then, AD=BD=12AB,BE=EC=12BC And AF=CF=12AC Now, using the mid-point theorem, EF || AB and EF=12AB=AD=BDED || AC and ED=12AC=AF=CF And DF || BC and DF=12BC=BE=CE In ΔADFandΔEFD, AD=EF AF=DE DF = FD [Common] ∴ΔADF≅ΔEFD [by SSS congruence rule] Similarly, ΔDEF≅ΔEDB And Δ≅ΔCFE So, ΔABC is divided into four congruent triangles. Hence proved.

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