Math, asked by Anonymous, 7 months ago

Explain Thales theorem and prove it ...


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Answered by Anonymous
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Fig :- Thales theorem

Thales theorem :-

According to the Thales Theorem, if a line is drawn || to one side of a triangle, to interest the other two sides at indistinct points, the other two sides are divided in the same ratio. Thales theorem is also known as Basic Proportionality Theorem (B.P.T.).

✏ Thales theorem Proof :-

Given : - In △ABC,DE∥BC

To prove :- \dfrac{AD}{DB}=\dfrac{AE}{EC}

Construction :- Join BE and CD . Draw EF⊥AB and DN ⊥ AL .

Proof :-

ar(△ADE) = \dfrac{1}{2}×AD×EF

= \dfrac{1}{2}×AE×DN

▶ar(△BDE) = \dfrac{1}{2} ×BD×EF

▶ar(△CDE) = \dfrac{1}{2}×EC×DN

▶ar(△ADE)/ar(△BDE) = \dfrac{1}{2} ×AD×EF/\dfrac{1}{2}×BD×EF ........(1)

▶ ar(△ADE)/ar(△CDE) = \dfrac{1}{2}×AE×DN/\dfrac{1}{2}×EC×DN ........(2)

▶But, ar(△BDE) = ar(△CDE) ..............(3)

As they are on the same base DE and DE ∥ BC

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

➡ AD/DB = AE/EC

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Answered by thapaavinitika6765
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