# probability based on geometry

A rectangle is drawn where the lengths of the sides are chosen randomly from [0, 10] and independently of one another. Find the probability that the length of its diagonal is smaller than or equal to 10.

Can I assume that the length of sides is a random variable having R[0,10] distribution?

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We can make things a lot simpler by consider a rectangle on the Cartesian plane.

The $4$ vertexes of the rectangle can be denoted as $(0,0)$, $(a,0)$, $(b,0)$, $(a, b$), where $0\leq a, b\leq 10$.

Then, the question becomes to find the probability that point $(a, b)$ lies inside the circle with radius $10$, centered at $(0,0)$.

Can you get it from here?

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I think I can.Thank You. – kris91 May 8 '13 at 14:02
So basically I have to find P($a^2$ + $b^2$<100) – kris91 May 8 '13 at 14:12
yes you have to find $P[a^2+b^2 < 100]$ for $a,b \in [0,10]$ – Uma kant May 8 '13 at 14:23

$$Prob. = \frac{Area ~ of ~(x^2+y^2 \leq 100,0 \leq x \leq 10,0 \leq y \leq 10)}{Area~ of~ rectangle ~ (0 \leq x \leq 10,0 \leq y \leq 10)} = \pi/4$$