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Byju's Answer
Standard IX
Mathematics
RHS Criteria for Congruency
In an isoscel...
Question
In an isosceles triangle with base a and a lateral side b the vertex angle is equal to
20
∘
. Prove that
|
a
|
3
+
|
b
|
3
=
3
|
a
|
|
b
|
2
.
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Solution
A
C
=
a
A
B
=
B
C
=
b
Let
M
∑
B
C
such that
A
M
=
a
Hence
∠
B
A
M
=
80
0
−
20
0
=
60
0
Now let
N
∑
A
B
such that
A
N
=
a
Here
A
N
=
A
M
=
M
N
=
a
and
∠
N
M
B
=
180
0
−
80
0
−
60
0
=
40
0
Now let
k
∑
B
C
such that
N
k
=
a
Then,
∠
N
k
M
=
40
0
which says that
∠
B
N
k
=
20
0
and from here
D
k
=
a
Now
k
L
D
altitude of
Δ
B
N
k
and
M
F
D
altitude of
Δ
A
M
C
Here since
B
N
=
b
−
a
and
Δ
B
L
k
≅
Δ
A
F
M
we obtain
A
F
=
B
C
=
b
−
a
2
and
F
C
=
a
−
b
−
a
2
=
3
a
−
b
2
Also we see that
Δ
A
B
C
∼
Δ
A
C
M
which gives
M
C
a
=
b
a
or
M
C
=
b
2
a
Since
A
M
2
−
A
F
2
=
M
C
2
−
F
C
2
we obtain
a
3
+
b
3
=
3
a
b
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