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Byju's Answer
Standard XII
Mathematics
Orthocentre
The vertices ...
Question
The vertices of a
△
ABC are A(5, 5) B(1, 5) and C(9, 1). A line is drawn to intersect sides AB and AC at p and Q respectively, such that
A
P
A
B
=
A
Q
A
C
=
3
4
. Find the length of the line segment PQ.
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Solution
We have,
A
P
A
B
=
A
Q
A
C
=
3
4
⇒
A
P
A
P
+
P
B
=
A
Q
A
Q
+
Q
C
=
3
4
⇒
A
P
A
P
+
P
B
=
3
4
,
A
Q
A
Q
+
Q
C
=
3
4
⇒
4
A
P
=
3
A
P
+
3
P
B
a
n
d
4
A
Q
=
3
A
Q
+
3
Q
C
⇒
A
P
=
3
P
B
a
n
d
A
Q
=
3
Q
C
⇒
A
P
P
B
=
3
1
a
n
d
A
Q
Q
C
=
3
1
⇒
P
a
n
d
Q
d
i
v
i
d
e
A
B
a
n
d
A
C
r
e
s
p
e
c
t
i
v
e
l
y
i
n
t
h
e
s
a
m
e
r
a
t
i
o
3
:
1
Thus, the coordinates of P and Q are
(
3
×
1
+
1
×
5
3
+
1
,
3
×
5
+
1
×
5
3
+
1
)
=
(
2
,
5
)
a
n
d
(
3
×
9
+
1
×
5
3
+
1
,
3
×
1
+
1
×
5
3
+
1
)
=
(
8
,
2
)
∴
P
Q
=
√
(
2
−
8
)
2
+
(
5
−
2
)
2
=
√
45
=
3
√
5
u
n
i
t
s
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