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Prove that, in a right triangle, the squareof the hypotenuse is equal to the sum ofthe squares of the other two sides.ICRSE 20181 |
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Answer» Given:A right angled ∆ABC, right angled at B To Prove- AC²=AB²+BC²Construction: draw perpendicular BD onto the side AC .Proof: We know that if a perpendicular is drawn from the vertex of a right angle of a right angled triangle to the hypotenuse, than triangles on both sides of the perpendicular are similar to the whole triangle and to each other. We have△ADB∼△ABC. (by AA similarity)Therefore, AD/ AB=AB/AC (In similar Triangles corresponding sides are proportional) AB²=AD×AC……..(1)Also, △BDC∼△ABCTherefore, CD/BC=BC/AC (in similar Triangles corresponding sides are proportional) Or, BC²=CD×AC……..(2)Adding the equations (1) and (2) we get,AB²+BC²=AD×AC+CD×ACAB²+BC²=AC(AD+CD)( From the figure AD + CD = AC)AB²+BC²=AC . ACTherefore, AC²=AB²+BC² This theroem is known as Pythagoras theroem. |
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