9.
In AABC, if
3
1
atc btc a+b+c
then prove that C = 60°
Answers
Answered by
13
Answer:
ANSWER
Formula:
c
2
=a
+
b
2
−2abcosC
Given, ∠C=60
∘
∴c
2
=a
+
b
2
−2abcos60
c
2
=a
+
b
2
−ab..............(1)
To prove:
a+c
1
+
b+c
1
=
a+b+c
3
a+c
1
+
b+c
1
=
a+b+c
3
(a+c)(b+c)
a+b+2c
=
a+b+c
3
(a+b+2c)(a+b+c)=3[(a+c)(b+c)]
a
2
+ab+ac+ba+b
2
+bc+2ca+2cb+2c
2
=3ab+3ac+3cb+3c
2
a
2
+b
2
+2ab−3ab=3c
2
−2c
2
a
2
+b
2
−ab=c
2
.............(2)
From (1) and (2) it's clear that,
a+c
1
+
b+c
1
=
a+b+c
3
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