Let the interval [-r,r] be the base of a semicircle. If a point is selected at random from this interval, assign a probability to the event that the length of the perpendicular segment from the point to the semicircle is less than r/2.
Let the interval [-r,r] be the base of a semicircle. If a point is selected at random from this interval, assign a probability to the event that the length of the perpendicular segment from the point to the semicircle is less than r/2.
You can use Pythagoras' theorem to calculate the fraction of the diameter for which the vertical line
to the semicircle is under half the radius.
This is the base length of two right-angled triangles both sides of the y-axis.
You only need to work with one of them.
To calculate the length of the line segment for which the perpendicular is under half the radius...
The probability is the ratio of that length to the radius.
The probability is the ratio of 2(r-x) to 2r,
because if the point is on either line segment of length r-x (on the diameter,
touching the circumference), then the perpendicular height to the semicircle
is less than half the radius.
You can also use trigonometry to find the ratio.