Question

What is Thales Theorem??Explain.

Answer 1

→Thales theorem is also known as Basic Proportionality 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 sided are divided in the same ratio.

Given,

In ∆ABC, DE || BC which intersects sides AB and AC at D and E respectively,

Required to prove;-

AD/DB = AE/EC

Construction:'

Join B,E and C,D and then draw DM ⊥ AC and EN ⊥ AB.

Proof:-

Area of ∆ADE = ½ × AD × EN

Area of ∆BDE = ½ × BD × EN

ar (∆ADE)/ar (∆BDE) = (½ × AD × EN)/(½ × BD × EN)

= AD/BD -------(1)

Area of ∆ADE = ½ × AE × DM

Area of ∆CDE = ½ × EC × DM

ar (∆ADE)/ar (∆CDE) = (½ × AE × DM)/(½ × EC × DM)

= AE/EC ------(2)

Observe that ∆BDE and ∆CDE are on the same base DE and between same parallels BC and DE,

ar (∆BDE) = ar (∆CDE) ------(3)

From (1) (2) and (3), we get

AD/DB = AE/EC

Hence proved.



Hope it helps