State and prove converse of Basic Proportionality theorem.
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Answers
Converse of Basic Proportionality Theorem. Converse of Basic Proportionality Theorem: If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side. Given : A Δ ABC and a line intersecting AB in D and AC in E, such that AD / DB = AE / EC.
Answer:
If a straight line is drawn parallel to one side of a triangle intersecting the other two sides then it divides the two sides in the same ratio.
Step-by-step explanation:
Let ABC be the triangle.
The line l parallel to BC intersect AB at D and AC at E.
To prove
DB
AD
=
EC
AE
Join BE,CD
Draw EF⊥AB, DG⊥CA
Since EF⊥AB,
EF is the height of triangles ADE and DBE
Area of △ADE=
2
1
× base × height=
2
1
AD×EF
Area of △DBE=
2
1
×DB×EF
areaofΔDBE
areaofΔADE
=
2
1
×DB×EF
2
1
×AD×EF
=
DB
AD
........(1)
Similarly,
areaofΔDCE
areaofΔADE
=
2
1
×EC×DG
2
1
×AE×DG
=
EC
AE
......(2)
But ΔDBE and ΔDCE are the same base DE and between the same parallel straight line BC and DE.
Area of ΔDBE= area of ΔDCE ....(3)
From (1), (2) and (3), we have
DB
AD
=
EC
AE