Prove that the probability that two positive numbers, x and y both less than 1, written down at random with unity, yield a triplet(x,y,1) whish are the sides of an obtuse angled triangle is $\displaystyle \frac{\pi - 2}{4}$
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Malay
Prove that the probability that two positive numbers, x and y both less than 1, written down at random with unity, yield a triplet(x,y,1) whish are the sides of an obtuse angled triangle is $\displaystyle \frac{\pi - 2}{4}$
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Malay
Have you considered revising you signature to :Originally Posted by ThePerfectHacker
1) do not make posts of cardinality greater than 1
2) do not discuss illegal operations on the forum
3) do not use improper integrals
4) do not abuse notation
5) Axiom of Choice is allowed.
RonL
Here's the picture. The probability is the area between the thicker lines.Originally Posted by malaygoel
$\displaystyle \setlength{\unitlength}{2.5cm}
\begin{picture}(1,1)
\qbezier(0,0)(0,0)(1,0)
\qbezier(0,0)(0,0)(0,1)
\qbezier(1,0)(1,0)(1,1)
\qbezier(0,1)(0,1)(1,1)
\linethickness{1.3pt}
\qbezier(1,0)(1,0)(0,1)
\qbezier(1.0, 0.0)(1.0, 0.41)(0.71, 0.71)
\qbezier(0.71, 0.71)(0.41, 1.0)(0.0, 1.0)
\end{picture}
$
Ok, I understand it.Originally Posted by malaygoel
I drew the graphs of
$\displaystyle x + y = 1$
$\displaystyle x^2 + y^2 = 1$
and got the same picture as yours.
I got the answer also from the picture.
Did you also arrive at the picture using Analytic Geometry?If no, what method you adopted.
Thanks for your help,JakeD
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Malay
The inequalities involving $\displaystyle x$ and $\displaystyle y$ come from considering what it means to be an obtuse triangle. That comes from geometry. But the picture comes from graphing those inequalities. Then the leap to the probability comes from seeing that the picture is of a unit circle and a triangle. That analytical geometry and trigonometry is not necessary. As I said you can use calculus there. But then the trig comes back in solving the integral for the circle. The answer contains $\displaystyle \pi$ remember.Originally Posted by malaygoel
We have to draw arough sketch, we can't use graphs.Originally Posted by JakeD
Trigonometry can be used as extensively as you want.$\displaystyle \pi$ can arise from trigonometry.Then the leap to the probability comes from seeing that the picture is of a unit circle and a triangle. That analytical geometry and trigonometry is not necessary. As I said you can use calculus there. But then the trig comes back in solving the integral for the circle. The answer contains $\displaystyle \pi$ remember.
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Malay