+2115 Triangle XYZ is equilateral, with O as its center. A point P is chosen at random. Find the probability that P is closer to point O than to any of the side lengths.
+121 Here is my attempt
To find the probability that point P is closer to point O than to any of the side lengths in equilateral triangle XYZ, we need to understand the geometry involved. Here are the steps to solve this problem:
1. Draw equilateral triangle XYZ with center O.
2. The region closer to O than any of the side lengths forms an equilateral triangle inside triangle XYZ. Let's call this inner equilateral triangle LMN.
3. The area closer to the side lengths than to O consists of three smaller equilateral triangles that are outside the inner equilateral triangle.
4. Calculate the ratio of the area of the inner equilateral triangle to the total area of triangle XYZ. This ratio will give us the probability that point P is closer to O than to any of the side lengths.
5. The ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding sides. Since the side length of the inner equilateral triangle is half that of triangle XYZ, the area ratio is (1/2)^2 = 1/4.
6. Therefore, the probability that a randomly chosen point P is closer to O than to any of the side lengths is 1/4 or 25%. This explanation should help you understand how to calculate the probability in this scenario.
