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Byju's Answer
Standard X
Mathematics
AAA Similarity
In an equilat...
Question
In an equilateral triangle
A
B
C
, the side
B
C
is trisected at
D
i.e.,
B
D
=
1
3
B
C
. Prove that
9
A
D
2
=
7
A
B
2
.
Open in App
Solution
G
i
v
e
n
:
△
A
B
C
i
s
a
n
e
q
u
i
l
a
t
e
r
a
l
t
r
i
a
n
g
l
e
,
w
h
o
s
e
s
i
d
e
B
C
i
s
t
r
i
s
e
c
t
e
d
a
t
D
.
T
o
P
r
o
v
e
:
9
A
D
2
=
7
A
B
2
P
r
o
o
f
:
△
A
B
C
b
e
a
n
e
q
u
i
l
a
t
e
r
a
l
t
r
i
a
n
g
l
e
a
n
d
D
b
e
t
h
e
p
o
i
n
t
o
n
B
C
s
u
c
h
t
h
a
t
B
D
=
1
3
B
C
(
G
i
v
e
n
)
D
r
a
w
A
E
⊥
B
C
,
J
o
i
n
A
D
.
B
E
=
E
C
(
A
l
t
i
t
u
d
e
d
r
a
w
n
f
r
o
m
a
n
y
v
e
r
t
e
x
o
f
a
n
e
q
u
i
l
a
t
e
r
a
l
t
r
i
a
n
g
l
e
b
i
s
e
c
t
s
t
h
e
o
p
p
o
s
i
t
e
s
i
d
e
)
S
o
,
B
E
=
E
C
=
B
C
2
I
n
△
A
B
C
A
B
2
=
A
E
2
+
E
B
2
.
.
.
(
1
)
A
D
2
=
A
E
2
+
E
D
2
.
.
.
(
2
)
F
r
o
m
(
1
)
a
n
d
(
2
)
A
B
2
=
A
D
2
−
E
D
2
+
E
B
2
⇒
A
B
2
=
A
D
2
B
C
2
36
+
B
C
2
4
(
∵
B
D
+
D
E
=
B
C
2
⇒
B
C
3
+
D
E
=
B
C
2
⇒
D
E
=
B
C
6
)
⇒
A
B
2
+
B
C
2
36
−
B
C
2
4
=
A
D
2
(
∴
E
B
=
B
C
2
)
⇒
A
B
2
+
A
B
2
36
A
B
2
4
=
A
D
2
(
∴
A
B
=
B
C
)
⇒
36
A
B
2
+
A
B
2
−
9
A
B
2
36
=
A
D
2
⇒
28
A
B
2
36
=
A
D
2
⇒
7
A
B
2
=
9
A
D
2
H
e
n
c
e
P
r
o
v
e
d
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Similar questions
Q.
In an equilateral triangle ABC, D is a point on side BC such that BD =
BC. Prove that 9 AD
2
= 7 AB
2
.