Leg from a random length
I have a line segment of length 7 cm.
I split it at a random point to create two smaller segments. I take the the larger of the two segments and use it as the hypotenuse of a right triangle, and use the shorter segment as one of the legs.
What is the chance the other leg has a length of at least 5 cm?
Express your solution as a decimal, to 4 places.
The answer is 0.4898.
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1 solution
Let the hypotenuse be length x cm. Then the known leg is 7 − x cm, so the other leg is ( x 2 − ( 7 − x ) 2 ) 0 . 5 cm, which simplifies to ( 1 4 x − 4 9 ) 0 . 5 cm.
14x - 49 therefore has to be at least 25, so x is at least 37/7 cm.
That leaves two intervals of length 12/7 cm on either end of the segment where the split can happen, for a total chance of 24/49.