Question : A tree of height $h$ metres is broken by storm in such a way that its top touches the ground at a distance of $x$ metres from its root. Find the height at which the tree is broken. (Here $h>x$)
Option 1: $\frac{h^2+x^2}{2h}$ metre
Option 2: $\frac{h^2-x^2}{2h}$ metre
Option 3: $\frac{h^2+x^2}{4h}$ metre
Option 4: $\frac{h^2-x^2}{4h}$ metre
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Correct Answer: $\frac{h^2-x^2}{2h}$ metre
Solution :
Given:
AB = Height of tree = $h$ metre
Let the height at which the tree is broken, AC = $y$ metre
BC = CD = Broken part of tree = $(h – y)$ metre
∴ In ∆ ACD,
AC
2
+ AD
2
= CD
2
⇒ $y^2 + x^2 = (h – y)^2$
⇒ $y^2 + x^2 = h^2 + y^2 – 2hy$
⇒ $x^2 = h^2 – 2hy$
⇒ $2hy = h^2 – x^2$
$\therefore y =\frac{h^2-x^2}{2h}$ metre
Hence, the correct answer is $\frac{h^2-x^2}{2h}$ metre.
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