if a line is drawn parallel toone side of a a triangle to intersect the other twi side s in distinct points, the other two sides are divided in same ratio .proof
Answers
Step-by-step explanation:
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Theorem:
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points , then the other two sides are divided in the same ratio.
[ Basic Proportionality Theorem Or Thales Theorem ]
Given:
In ∆ABC, which intersects AB and AC at D and F respectively.
RTP:
AD / DB = AE / EC
Construction:
Join B, E and C, D and then draw
Proof:
Area of ∆ADE = 1 / 2 × AD × EN
Area of ∆BDE = 1 / 2 × BD × EN
So, ar( ∆ADE ) / ar( ∆BDE )
= ( 1 / 2 × AD × EN ) / ( 1 / 2 × BD × EN )
= AD / AB ----(1)
Again Area of ∆ADE = 1 / 2 × AE × DM
Area of ∆CDE = 1 / 2 × EC × DM
So, ar(∆ADE) / ar(∆CDE)
= ( 1 / 2 × AE × DM ) / ( 1 / 2 × EC × DM )
= AE / EC ------(2)
Observe that ∆BDE and ∆CDE are on the same base DE and between the same parallels BC and DE.
So, ar(∆BDE) = ar(∆CDE) ---(3)
From (1),(2) & (3), we have
= AD / DB = AE / EC
Hence, proved.
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