A point S is chosen inside the square MNPQ. What is the probability that the angle MSN is acute?
Draw square MNPQ.
Find the midpoint of MN, call it "X".
Using X as the center, draw the semicirce inside the circle that starts at M and ends at N.
Choose a point Y on the semicircle. Angle(MYN) will be a right angle because it is inscribed
in a semicircle.
If you choose a point Y' inside the semicircle, angle(MY'N) will be an obtuse angle.
If the point that you choose is outside the semicircle, it will be an acute angle.
Now, to find the probability of its being acute.
We need to compare the area outside the semicircle to the area of the whole square.
Assume that the square has a side length of 2.
Then, the area of the square is 4.
The area inside the semicircle will be: ½·pi·r2 = ½·pi·(1)2 = ½·pi.
The area outside the semicircle will be: 4 - ½·pi.
The probability will be the area outside the semicircle divided by the area of the square:
probability = ( 4 - ½·pi ) / 4
[Because we're finding a ratio, we can assume whatever we wish for the side length of
the square. If you are in doubt, choose another number for this length, and recalculate.]