two isosceles triangles whose vertical angles are equal are placed so as to have their vertices coincide. prove that the line joining their angular points are equal
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Let △ABC and △DEF be the given triangles such that AB=AC and DE=DF, ∠A=∠D
and
Area(△DEF)
Area(△ABC)
=
25
16
.......(i)
Draw AL⊥BC and DM⊥EF.
Now, AB=AC,DE=DF
⇒
AC
AB
=1 and
DF
DE
=1
⇒
AC
AB
=
DF
DE
⇒
DE
AB
=
DF
AC
Thus, in triangles ABC and DEF, we have
DE
AB
=
DF
AC
and ∠A=∠D [Given]
So, by SAS-similarity criterion, we have
△ABC∼△DEF
⇒
Area(△DEF)
Area(△ABC)
=
DM
2
AL
2
⇒
25
16
=
DM
2
AL
2
[Using (i)]
⇒
DM
AL
=
5
4
Hence, AL:DM=4:5
Step-by-step explanation:
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