A geometry problem by Hahn Lheem

Geometry Level 3

Two points are picked at random on the circumference of a circle with equation $x^2+y^2=1$ . The probability that the length of the chord connecting these two points is greater than $\sqrt{2-\sqrt{2}}$ can be expressed as $\dfrac{m}{n}$ , where $m$ and $n$ are positive, coprime integers. Find $m^3+n^3$ .

The answer is 91.

1 solution

Ronak Agarwal
Jun 2, 2014

Distance of the chord is greater than (2-2^(0.5))^(0.5) implies the angle between the two points is greater than 45 degrees hence the range of angle is 45<theta<325.Hence the probability is given by (upper limit-lower limit)/(total angle)=(325-45)/360 = 3/4. So m=3,n=4. m^3+n^3=91

Did you mean 315 instead of 325?

Anand Iyer - 6 years, 5 months ago

A geometry problem by Hahn Lheem

The answer is 91.

1 solution

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