1 Theorem 6.9- In a triangle, if one side is equal to the sum.JPG

2 Theorem 6.9- In PQR, By pythagoras theorem.JPG
3 Theorem 6.9-Now in ABC and PQR.JPG

  1. Chapter 6 Class 10 Triangles
  2. Serial order wise
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Transcript

Theorem 6.9 :- In a right triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle. Given :- A triangle ABC in which 𝐴𝐶﷮2﷯= 𝐴𝐵﷮2﷯+ 𝐵𝐶﷮2﷯ To Prove :- ∠B=90° Construction :- Draw Δ PQR right angled at Q, such that PQ = AB and QR = BC. Proof :- In ∆PQR ∠Q=90° By Pythagoras theorem, 𝑃𝑅﷮2﷯= 𝑃𝑄﷮2﷯+ 𝑄𝑅﷮2﷯ Since PQ = AB and QR = BC ⇒ 𝑃𝑅﷮2﷯= 𝐴𝐵﷮2﷯+ 𝐵𝐶﷮2﷯ Also, given that 𝐴𝐶﷮2﷯= 𝐴𝐵﷮2﷯+ 𝐵𝐶﷮2﷯ From (1) & (2) 𝑃𝑅﷮2﷯= 𝐴𝐶﷮2﷯ PR = AC Now, In Δ ABC & Δ PQR AC = PR AB = PQ BC = QR ∴ Δ ABC ≅ Δ PQR ⇒ ∠ B = ∠ Q Since ∠ Q = 90° ∠ B = 90° Hence Proved.

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