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Byju's Answer
Standard IX
Mathematics
Equal Intercept Theorem
AB is a line ...
Question
A
B
is a line segment
P
and
Q
are point on opposite sides of
A
B
such that each of them is equidistant from the points
A
and
B
. Show that the line
P
Q
is perpendicular bisector of
A
B
.
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Solution
Given
P
is equidistant from points
A
and
B
P
A
=
P
B
.....
(
1
)
and
Q
is equidistant from points
A
and
B
Q
A
=
Q
B
.....
(
2
)
In
△
P
A
Q
and
△
P
B
Q
A
P
=
B
P
from
(
1
)
A
Q
=
B
Q
from
(
2
)
P
Q
=
P
Q
(common)
So,
△
P
A
Q
≅
△
P
B
Q
(SSS congruence)
Hence
∠
A
P
Q
=
∠
B
P
Q
by CPCT
In
△
P
A
C
and
△
P
B
C
A
P
=
B
P
from
(
1
)
∠
A
P
C
=
∠
B
P
C
from
(
3
)
P
C
=
P
C
(common)
△
P
A
C
≅
△
P
B
C
(SAS congruence)
∴
A
C
=
B
C
by CPCT
and
∠
A
C
P
=
∠
B
C
P
by CPCT ....
(
4
)
Since,
A
B
is a line segment,
∠
A
C
P
+
∠
B
C
P
=
180
∘
(linear pair)
∠
A
C
P
+
∠
A
C
P
=
180
∘
from
(
4
)
2
∠
A
C
P
=
180
∘
∠
A
C
P
=
180
∘
2
=
90
∘
Thus,
A
C
=
B
C
and
∠
A
C
P
=
∠
B
C
P
=
90
∘
∴
,
P
Q
is perpendicular bisector of
A
B
.
Hence proved.
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Similar questions
Q.
AB is a line segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (See Figure). Show that the line PQ is perpendicular bisector of AB.