10. In Fig. D is the mid-point of side BC and AE I BC. If BC= a AC= b. AB= C, ED= x, AD = p and
AE = h, prove that: (b2-c2) = 2ax
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Answer:
D is the midpoint of side BC and AE is perpendicular to BC
Answer
Given:
AE⊥BC
BD=CD
BC=a,AB=c,AD=p,AC=b,ED=x,AE=h
Proof:-
In △ABC,
AB
2
=AE
2
+BE
2
⇒AB
2
=AE
2
+(BC−ED−CD)
2
⇒AB
2
=AE
2
+(BC−ED−
2
BC
)
2
⇒c
2
=h
2
+(
2
a
−x)
2
⇒c
2
=h
2
+
4
a
2
+x
2
−ax-------------(1)
In △AEC,
AC
2
=AE
2
+CE
2
⇒b
2
=h
2
+(
2
BC
+ED)
2
⇒b
2
=h
2
+(
2
a
+x)
2
⇒b
2
=h
2
+
4
a
2
+x
2
+ax---------------(2)
In △AED
AE
2
+ED
2
=AD
2
⇒h
2
+x
2
=p
2
-------------------(3)
Using Equation (3) in (1), we get
c
2
=h
2
+x
2
+
4
a
2
−ax
c
2
=p
2
−ax+
4
a
2
-----------------------(4)
Using Equation(3) in (2),
b
2
=h
2
+x
2
+
4+ax
a
2
b
2
=p
2
+ax+
4
a
2
-----------------------(5)
Adding Equation (4) & (5), we get
c
2
+b
2
=2p
2
+
2
a
2
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