Question

Prove that the midsegments of any triangle divide it into four congruent triangles

Answer 1

Answer:

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

Based on the midpoint theorem

EF= ½ BC

In the same way

FD= ½ AC and ED= ½ AB

Consider △AFE and △BFD

consider △AFE and △BFD

We know that AF=FB

Based on the midpoint theorem

FE= ½BC=BD

FD= ½ 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.

hope this helps u ☺️✌