State and prove "In a right triangle,the square of hypotenuse is equal to the square of other two sides.
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State and prove "In a right triangle,the square of hypotenuse is equal to the square of other two sides.
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Answer:
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∼△ABC
Therefore, 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×AC
AB²+BC²=AC(AD+CD)
( From the figure AD+CD=AC)
AB²+BC²=AC.AC
Therefore, AC²=AB²+BC²
This theorem is known as Pythagoras theorem...