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Prove That converse Pythagoras Therom And With Figure also |
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Answer» Answer: a2 + b2 = C2 PROOF of Pythagoras Theorem Given: A ∆ XYZ in which ∠XYZ = 90°. To prove: XZ² = XY² + YZ² Construction: Draw YO ⊥ XZ Proof: In ∆XOY and ∆XYZ, we have, ∠X = ∠X → common ∠XOY = ∠XYZ → each equal to 90° Therefore, ∆ XOY ~ ∆ XYZ → by AA-similarity ⇒ XO/XY = XY/XZ ⇒ XO × XZ = XY²----------------- (i) In ∆YOZ and ∆XYZ, we have, ∠Z = ∠Z → common ∠YOZ = ∠XYZ → each equal to 90° Therefore, ∆ YOZ ~ ∆ XYZ → by AA-similarity ⇒ OZ/YZ = YZ/XZ ⇒ OZ × XZ = YZ² ----------------- (ii) From (i) and (ii) we GET, XO × XZ + OZ × XZ = (XY² + YZ²) ⇒ (XO + OZ) × XZ = (XY² + YZ²) ⇒ XZ × XZ = (XY² + YZ²) ⇒ XZ² = (XY² + YZ²) |
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