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If ‘A’ is the area of a right triangle and ‘b’ is one of the sides containing the right angle ,then prove that the length of the altitude on the hypotenuse is 2Ab/squrare root of b^2+4A^2

If ‘A’ is the area of a right triangle and ‘b’ is one of the sides containing the right angle ,then prove that the length of the altitude on the hypotenuse is 2Ab/squrare root of b^2+4A^2

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2 Answers

Ajay Verma
askIITians Faculty 33 Points
9 years ago
soln:203-125_01.png
altitude = d

area = ½ a*b =A = ½ d * (AB) …....................(1)

AB = (a2+b2)1/2
= ( 4A2/b2+b2)1/2

putting value on eqn1
A =½ d *( 4A2/b2 +b2)1/2

after solving
d = 2Ab / ( 4A2+ b2)1/2


Regards
Sumit Majumdar IIT Delhi
askIITians Faculty 137 Points
9 years ago
Dear student,
Let us assume a right angled triable ABC with the right angle being at B.
Let AB = b. Also let BC = c.
68-405_1.jpg
Hence we get:
c=\frac{2A}{b}
Also, using the Pythagoras theorem, we get:
AC=\frac{\sqrt{b^{4}+4a^{2}}}{b}
Hence, finding the area we get:
\frac{1}{2}\frac{\sqrt{b^{4}+4a^{2}}}{b}BD=A\Rightarrow BD=\frac{2bA}{\sqrt{b^{4}+4a^{2}}}
Regards
Sumit

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