Circling the probability
Probability
Level
5
The maximum length (x) possible for a chord of a circle is 2r (where r is the radius of the circle ).Of all the possible values of x what is the probability of x lying between 1/2 and 5/6 of 2r ?
Give your answer to 2 decimal places.
The answer is 0.45.
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1 solution
Consider a chord AB (of length x) of a circle with centre O and radius r. Let the length of the line joining the midpoint M of the chord AB and the centre O be 'a' units.
Let θ be the angle between OM and OB (or OA). sinθ = x/2r.........(1) cosθ = a/r.........(2)
According to the question : (2r)/2 < x < (2r)*5/6
r < x< 5r/3
r < 2rsinθ < 5r/3....from (1)
1< 2sinθ <5/3
1< 4sin^2θ< 25/9
1< 4-4cos^2(θ) < 25/9
3 /4> cos^2(θ) > 11/9
sqrt(3)/2 >cosθ >sqrt(11) /6
multiplying each term by r.
(r)sqrt(3)/2 >rcosθ >(r)sqrt(11) /6
(r)sqrt(3)/2 >a>(r)sqrt(11) /6......from(2)
For this condition to be satisfied the point M has to lie in the region between concentric circles of radius (r)sqrt(11) /6 and (r)sqrt(3)/2 .
Probability that length of a chosen chord of a circle lies between 1/2 and 5/6 of its diameter
=area of the region between the circles / area of bigger circle
= 4/9 (approximately 0.45)