what types of triangle express
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In an acute-angled triangle, express a median in terms of its sides.
Hard
Solution
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Given:-
In △ABC,AD is the median to the side BC.
Construction: AE⊥BC
BD=CD=
2
1
BC (∵ D is the mid-point of the side BC)--------(1)
In △AED,
AD
2
=AE
2
+DE
2
(Pythagoras Theorem.)
⇒AE
2
=AD
2
−DE
2
------------(2)
In △AEB,
AB
2
=AE
2
+BE
2
⇒AB
2
=AD
2
−DE
2
+BE
2
----------------{From Equation(2)}
⇒AB
2
=AD
2
−DE
2
+(BD+DE)
2
(∵BE=BD+DE)
⇒AB
2
=AD
2
−DE
2
+BD
2
+DE
2
+2BD⋅DE
⇒AB
2
=AD
2
+BD
2
+2BD⋅DE
⇒AB
2
=AD
2
+
4
BC
2
+2⋅
2
1
⋅BC⋅DE (From Equation(1))
⇒AB
2
=AD
2
+
4
BC
2
+BC⋅DE----------(3)
In △AED,
AC
2
=AE
2
+EC
2
=AD
2
−DE
2
+EC
2
=AD
2
−DE
2
+(DC−DE)
2
=AD
2
−DE
2
+DC
2
+DE
2
−2DC⋅DE
=AD
2
+DC
2
−2DC×DE
=AD
2
+(
2
1
BC)
2
−2(
2
1
BC)(DE)
=AD
2
+
4
BC
2
−BC⋅DE------------(4)
Adding Equation (3) & (4), we get
AB
2
+AC
2
=
4
1
BC
2
+AD
2
+BC⋅DE+AD
2
+
4
1
BC
2
−BC⋅DE
⇒(AB
2
+AC
2
)=
2
BC
2
+2AD
2
⇒2AB
2
+2AC
2
=BC
2
+4AD
2
∴ The relationship between the sides of a triangle and its median is given by
2AB
2
+2AC
2
=BC
2
+4AD
2
solution