# Point in the square

In $$1 \times 1$$ randomly select a point. Let $$x > 0$$. Find the probability of:

1. The distance between the point and fixed side of the square is not more than $$x$$.
2. The distance between the point and closest side of the square is not more than $$x$$.
3. The distance between the point and diagonals is not more than $$x$$.

As I understand, the answer for the first question is the relation of area of $$1 \times x$$ rectangle to the area of square. That is, $$\frac{1 \cdot x}{1} = x$$. I can't understand what changes in the second question though and where to start in the third question.

In the first question, the distance between the point and fixed side is not more than $$x$$ in the rectangle $$1\times x$$ near this fixed side. And in the second question, the distance is not more than $$x$$ in every $$1\times x$$ rectangle adjacent to the side. So, the probability is $$\frac{1-(1-2x)^2}{1}$$ (Draw it and you'll see).
In the third, you should calculate the area of two $$x$$-wide layers adjacent to the diagonal on both sides (it obviously should not go beyond your $$1\times 1$$ square).
• Can you check the answer for the second question please? Mine is $\frac{1-(1-2x)^2}{1}$. Is that correct? Oct 7, 2019 at 23:33