Question

prove that the line segement joining the middle points of the side of triangle divide it into four conguent triangles
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Answer 1

From the figure we know that F and E are the midpoints of AB and AC

Based on the midpoint theorem

EF=

2

1

BC

In the same way

FD=

2

1

AC andED=

2

1

AB

Consider △AFE and △BFD

consider △AFE and △BFD

We know that AF=FB

Based on the midpoint theorem

FE=

2

1

BC=BD

FD=

2

1

AC=AE

By SSS congruence criterion

△AFE≅△BFD

Consider △BFD and △BFD

Consider △BFD and △FED

We know that FE≅BC

So we get FE≅BD andAB≅ED

Using the midpoint theorem

FB≅ED

Hence, □BDEF is a parallelogram

So we know that FD is a diagonal which divides the parallelogram into two congruent triangles

△BFD≅△FED

In the same way we can prove thatFECD is a parallelogram

△FED≅△EDC

So we know that △BFD,△FDE,△FED and △EDC are congruent to each other

Therefore, it is proved that the line segments joining the middle points of the sides of a triangle divide it into four congruent triangles.