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Prove that the four triangles formed by joining in pairs the mid-points of the sides C of a triangle are

congruent to each other

Answer:

It is given that

In Δ ABC

D, E, and F are the mid-points of AB, BC, and CA

Now join DE, EF, and FD

To find:

Δ ADF ≅ Δ DBE ≅ Δ ECF ≅ Δ DEF

To prove:

In Δ ABC

D and E are the mid-points of AB and BC

DE || AC or FC

Similarly DF || EC

DECF is a parallelogram

We know that

Diagonal FE divides the parallelogram DECF in two congruent triangles DEF and CEF

Δ DEF ≅ Δ ECF …… (1)

Here we can prove that

Δ DBE ≅ Δ DEF …. (2)

Δ DEF ≅ Δ ADF ……. (3)

Using equation (1), (2) and (3)

Δ ADF ≅ Δ DBE ≅ Δ ECF ≅ Δ DEF

"hey guys welcome to lido q a video
and we need your little tutor bringing
you this question on your screen
prove that the four triangles formed by
joining in pairs
the midpoints of the sides of a triangle
are congruent to each other so let us
see what the question means
right we draw a triangle
yeah now
after we draw the triangle let us mark
the midpoints
and join them
so this is given to us
right so a
b c d
e n f we have to prove
that triangle a
d e is congruent to triangle
b d f is congruent to triangle
f e c is congruent to triangle
d e f
how do we do that now
in the big triangle
right
d and e are
midpoints
of a b and ac
respectively
therefore by midpoint
formula
we can say that
this portion
ef right
so in this triangle ef is parallel to a
b
df is parallel to ac right
so ef is parallel to
a b or e d
and df
is parallel to ae
therefore we can say that
this portion a d
f e
right this portion
is a parallelogram
a d f e
is a parallelogram
now we know that
diagonal
of a
parallelogram
divides
[Music]
it into
two congruent triangles
[Music]
right therefore triangle
ade is congruent to triangle
d e f similarly
we can prove triangle
a d e right
sorry triangle ade is parallel
to this then we have
triangle
ade is parallel to triangle def
okay then also we have
triangle
the other four triangles are also
parallel like
congruent triangle ade is congruent to
triangle
what are the parallel okay triangle
sorry not triangle ade
but triangle
bdf is congruent to triangle
def and
triangle cef is also congruent to
triangle
def therefore
triangle ade is congruent to
triangle dbf is congruent to
triangle def
is congruent to triangle f
e c isn't that easy guys right
so if you have a doubt please leave a
comment below do like the video and
subscribe to our channel i'll see you in
our next video
until then bye guys keep practicing"

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