Q. 254.2( 11 Votes )

Prove that in a r

Answer :

Let the given right - angled triangle be ΔABC, right - angled at B.



We need to prove AC2 = AB2 + BC2


Now let’s draw a perpendicular line BD to line AC


i.e., BDAC


Now we know,


If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.


Hence, ΔADB~ΔABC


Now applying the similar triangles condition, i.e., sides of similar triangles are same in ratio, we get



AD.AC = AB2………(i)


Now similarly consider the other set of similar triangles,


ΔBDC~ΔABC


Now applying the similar triangles condition, i.e., sides of similar triangles are same in ratio, we get



CD.AC = BC2………(ii)


Now adding equation (i) and (ii), we get


AD.AC + CD.AC = BC2 + AB2


Now taking AC common on LHS, we get


AC (AD + CD) = BC2 + AB2


Now from figure we can see that AD + CD = AC, so above equation becomes


AC(AC) = BC2 + AB2


AC2 = BC2 + AB2


Hence in a right - angled triangle square of the hypotenuse is equal to sum of the squares of the other two sides.


Hence Proved.


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