9. ABC is a triangle. D, E and F are the midpoints of AB, AC and BC re

1.	9. ABC is a triangle. D, E and F are the midpoints of AB, AC and BC respectively. Prove that DE andAF bisect each otherANSWERS
Answer» As D and E are mid-points of sides AB and BC of the triangle ABC, by Theorem 1, DE \|\| AC Similarly, DF \|\| BC and EF \|\| AB Therefore ADEF, BDFE and DFCE are all parallelograms. Now DE is a diagonal of the parallelogram BDFE, therefore, ∆ BDE ≅ ∆ FED Similarly ∆ DAF ≅ ∆ FED and ∆ EFC ≅ ∆ FED So, all the four triangles are congruent.

9. ABC is a triangle. D, E and F are the midpoints of AB, AC and BC respectively. Prove that DE andAF bisect each otherANSWERS

Answer»

As D and E are mid-points of sides AB and BC of the triangle ABC, by Theorem 1,

DE || AC

Similarly, DF || BC and EF || AB

Therefore ADEF, BDFE and DFCE are all parallelograms.

Now DE is a diagonal of the parallelogram BDFE,

therefore, ∆ BDE ≅ ∆ FED

Similarly ∆ DAF ≅ ∆ FED

and ∆ EFC ≅ ∆ FED

So, all the four triangles are congruent.