A floor is paved with tiles, each tile being a parallelogram such that the distance between pairs of opposite sides are $a$ and $b$ respectively, the length of diagonal being $L$. A stick of length $C$ falls on the floor parallel to the diagonal. Show that the probability that it will lie entirely on one tile is $$\left(1-\frac CL\right) ^2.$$
If a circle of diameter $d$ is thrown on the floor, show that the probability that it will lie on one tile is : $$\left(1-\frac da\right) \left(1-\frac db\right).$$ I tried this one hard. But unable to get the correct answer. What i did is - 1. Since the stick is to be parallel to the diagonal , it can lie within the parallelogram and shapes like hexagonal. 2. Since the altitude of the parallelogram is a and b. I assumed sides as x and y respectively. So, using area of parallelogram. 3. I ended with x = bk and y = ak. So the area of parallelogram comes out to be "abk". 4. But I am unable to find the ratio k.