SOLUTION: A point S is chosen inside the square MNPQ. What is the probability that the angle MSN is a right angle? Express your answer as a decimal to the nearest hundredths? A poi

The other tutor's answer to the second part is incorrect.
We draw the square:




The angle MSN will be a right angle if and only if the arc it subtends
has measure 180°, that is, the arc is a semicircle.

So we draw a circle with diameter MN:



S must be picked along the right side of the green circle, say like this:



in order for angle MSN to be a right angle.

So S must be picked as one of the points of a 1 dimensional semicircle out of
one of the points of a 2 dimensional area.  But a semicircle is just a curved
line and curved (or straight) lines have 0 thickness and thus 0 area.

So the answer is 0.

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A point S is chosen inside the square MNPQ.  What is the probability that the
angle MSN is acute? Express your answer as a decimal to the nearest hundredths?

Now this problem is not 0. For if point S chosen in this part of the area:



like this:



then angle MSN will necessarily be an acute angle.

So the desired probability is the area of this
figure:



divided by the area of the whole square


.

Suppose " r "  represents the radius of the green circle:



Then each side of the square MNPQ has length 2r, so its area is (2r)² or 4r².

The area of the green circle is pr2

So the semicircle that is cut away from the square in this figure



is one-half of that area or 

So the area of that figure is the area of the square minus the
area of the semicircle or 



That's the numerator of the desired probability.  The denominator is the
area of the square MNPQ or 4r², so the desired probability is

 =

 =

 =

 =

 = 0.6073009183

Rounded to tenths the probability is 0.6

So the probability that angle MSN is acute is about 0.6,
the probability that it is obtuse is about 0.4, and
the probability that it is a right angle is 0.  Seems
strange that that should be 0, since it is possible that
it could be a right angle, but the probability of an angle
being exactly 90° out of an infinite number of possibilities
is loss than 1 out of a trillion trillion! which could only be
0.

Edwin