A coin of radius 1 cm is tossed onto a plane surface that has been tesselated by right
triangles whose sides are 8 cm, 15 cm, and 17 cm long. What is the probability that the
coin lands within one of the triangles?
To find the probability that the coin lands within one of the triangles, we need to find the area of the region where the coin can land and divide it by the total area of the surface.
To find the area of the region where the coin can land, we need to consider the size of the triangles and compare it to the size of the coin.
First, let's find the area of one triangle using Heron's formula. The semi-perimeter of the triangle is calculated as:
s = (8 + 15 + 17) / 2 = 20
Now, we can use Heron's formula:
Area = sqrt(s * (s - 8) * (s - 15) * (s - 17))
Area = sqrt(20 * (20 - 8) * (20 - 15) * (20 - 17))
Area = sqrt(20 * 12 * 5 * 3)
Area = sqrt(3600)
Area = 60 cm²
The area of one triangle is 60 cm².
Now, let's find the area of the region where the coin can land. Since the coin has a radius of 1 cm, the area of the region where it can land will be a circle with a radius of 1 cm.
Area of the region where the coin can land = π * (1 cm)²
Area of the region where the coin can land = π cm²
The total area of the surface is the sum of the areas of all the triangles. Since the surface is tessellated by these triangles, we can assume there are an infinite number of them.
Total area of the surface = infinity * 60 cm²
Since the total area of the surface is infinite, the probability that the coin lands within one of the triangles is:
Probability = Area of the region where the coin can land / Total area of the surface
Probability = π cm² / (∞ * 60 cm²)
Probability = 0
Therefore, the probability that the coin lands within one of the triangles is 0.