Question

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D, E and F are respectively the mid-points of the sides AB, BC and AC of triangle ABC. Find the ratio of triangle DEF to triangle ABC.

✴ GIVE PROPER PROCEDURE ALONG WITH REASONS AND DIAGRAM.

❌ NO SPAMS ALLOWED ❌

Answer 1

Hye !!


good morning !!!


look you answer with attachment:-





simple method :-



we know that :-



area of (∆ABC )/ area of (∆DEF) = ( side of ∆ABC / Side of ∆DEF ) ²




USING THIS WE GET RESULT :-




HOPE IT HELPS :D



thanks

Answer 2

aloha!

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here is the answer with diagram:

=>given: D, E and F are mid points of sides AB, BC and AC.
=> to find: the ratio of ar ∆ DEF to the ar ∆ ABC.
=> solution:

we know when two mid points are joined in a triangle they become || to one of the side.
in the figure,
DF is parallel to BC.
DE is parallel to AC.
EF is parallel to AB.

thus we come to know that we get three parallelograms here:
◆ parallelogram DFEB
◆ parallelogram DFEC
◆ parallelogram DEFA

also, we know that the opposite angles of parallelogram are equal. so,

angle B = angle DFE------------(1)
angle C = angle EDF------------(2)
angle A = angle DEF------------(3)

by 1, 2 and 3 we can write:
∆ ABC is similar to ∆ DEF {by AAA similarly criteria.}

now,
as the line joining the mid points of a triangle is half the length of third side we can write:
DE is 1/2 of AC or AC= 2DE
EF is 1/2 of AB or AB= 2EF
DF is 1/2 of BC or BC= 2DF

we know that the ratio of area of two similar triangles is equal to the ratio of square of their sides. so,

area∆ DEF/ area∆ ABC = DF squared/ BC squared.

ar∆ DEF/ ar∆ ABC = DF squared/ 2DF squared

ar∆DEF/ ar∆ ABC = DF squared/ 4DF squared

ar∆DEF/ ar∆ ABC = 1/ 4

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hope it helps :^)