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Byju's Answer
Standard IX
Mathematics
The Mid-Point Theorem
E is the mid-...
Question
E
is the mid-point of the median
A
D
of
Δ
A
B
C
and
B
E
is produced to meet
A
C
at
F
.Show that
A
F
1
3
A
C
.
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Solution
Given: A
Δ
A
B
C
in which
E
is the mid point of median
A
B
and
B
E
is produced to meet
A
C
at
F
.
Refer image,
To prove:
A
F
=
1
3
A
C
Construction: Draw
D
G
|
|
B
F
intersecting
A
C
at
G
.
Proof: In
Δ
A
D
G
,
E
is the mid-point of
A
D
and
E
F
|
|
D
G
∴
A
F
=
F
G
........(1) [Converse of mid-point theorem]
In
Δ
F
B
C
,
D
is the midpoint if
B
C
and
D
G
|
|
B
F
.
∴
F
G
=
G
C
...........(2) [Converse of mid-point theorem]
From equation (1) and (2), we get,
A
F
=
F
G
=
G
C
.........(3)
But,
A
C
=
A
F
+
F
G
+
G
C
⇒
A
C
=
A
F
+
A
F
+
A
F
(Using (3))
⇒
A
C
=
3
A
F
⇒
A
F
=
1
3
A
C
Hence proved.
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Q.
E is the mid-point of a median AD of
Δ
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