(8) Prove
a line
of triangle
that if
is
The other 2 sides
same
drawn parallel to one side
dinected in
diagonals of treptus
are
ratio .
5 how
that
Using above
cut carn
other
the same ratio
Answers
Answer:
ģ
Step-by-step explanation:
ģ
ģof triangle
that if
is
The other 2 sides
same
drawn parallel to one side
dinected in
diagonals of treptus
are
ratio .
5 how
that
Using above
cut carn
other
the same ratio
Step-by-step explanation:
Both these triangles are on the same base AB and have equal height EQ.
A
r
e
a
o
f
A
D
E
A
r
e
a
o
f
B
D
E
=
1
2
×
A
D
×
E
Q
1
2
×
B
D
×
E
Q
A
r
e
a
o
f
A
D
E
A
r
e
a
o
f
B
D
E
=
A
D
B
D
Now consider triangles CDE and ADE.
BPT Traingles Image
Both these triangles are on the same base AC and have equal height DP.
A
r
e
a
o
f
A
D
E
A
r
e
a
o
f
C
D
E
=
1
2
×
A
E
×
D
P
1
2
×
C
E
×
D
P
A
r
e
a
o
f
A
D
E
A
r
e
a
o
f
C
D
E
=
A
E
C
E
Both the triangles BDE and CDE are between the same set of parallel lines.
A
r
e
a
o
f
t
r
a
i
n
g
l
e
B
D
E
=
A
r
e
a
o
f
t
r
a
i
n
g
l
e
C
D
E
Applying this we have:
A
r
e
a
o
f
t
r
a
i
n
g
l
e
A
D
E
A
r
e
a
o
f
t
r
a
i
n
g
l
e
B
D
E
=
A
r
e
a
o
f
t
r
a
i
n
g
l
e
A
D
E
A
r
e
a
o
f
t
r
a
i
n
g
l
e
C
D
E
A
D
B
D
=
A
E
C
E
Corollary:
The above proof is also helpful to prove another important theorem called the mid-point theorem.
The mid-point theorem states that a line segment drawn parallel to one side of a triangle and half of that side divides the other two sides at the midpoints.
Conclusion:
Hence we prove the Basic Proportionality Theorem.
Therefore the line DE drawn parallel to the side BC of triangle ABC divides the other two sides AB, AC in equal proportion.