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Question

# D, E, and F are respectively the mid-points of the sides AB, BC and CA of a ΔABC. Triangles are formed by joining these mid-points D, E and F. Which among the following is true?

A
ΔEFDΔCFE
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B
ΔDEFΔEDB
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C
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D
All the above
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Solution

## The correct option is D All the aboveGiven: In a ΔABC, D, E and F respectively the mid-points of the sides AB, BC and CA. 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 And, DF = FD [Common] ∴ ΔADF≅ΔEFD [by SSS congruence rule] Similarly, ΔDEF≅ΔEDB And, ΔEFD≅ΔCFE So, ΔABC is divided into four congruent triangles. Hence proved.

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