Problem number 3 states, a certain park is shaped like an equilateral triangle. A cat wants to walk along one of the three indicated paths, thicker lines, from the upper corner to the lower right corner. The lengths of the paths are P, Q, and R as shown. Which of the following statements about the lengths of the path is true? So these are the paths that the cat took and we want to find the relative relationships between them. So first, let us consider this bottom edge. Since the cat took this bottom edge of the equilateral triangle on all three paths, this means that we can actually completely disregard this edge, because it does not matter if the cat actually traveled on this edge, it would not change the relative lengths. Now let's consider the sides of the path in green. Since the intermediary paths in yellow are perfectly horizontal, we know the ending height of one segment of the side paths is the starting height of another segment. This means that the sum of the side paths is equal to the side length of the triangle for each of the triangles. Since this is all the same, we can again disregard it. At this point, a large part of the path is disregarded, and we are just left with the intermediary paths that go through the middle of the park. Since the lines are perfectly horizontal, we know that the lower the path is on the triangle, the longer it is, and the higher the path is, the shorter it is. So here we have three different lengths of intermediary paths, long in magenta, medium in yellow, and short in orange. Let's call their lengths a, b, and c, and we know their relative lengths based on how high they are on the triangle, so a is less than b, which is less than c. We know that p equals a plus b, q equals b plus c, and r equals c plus a. At this point, it might be helpful to replace a, b, and c with dummy numbers to figure out the relationship, but instead we can see that b and c are the biggest numbers. So we must know that their sum will be the greatest, and a, b are the smallest numbers, so their sum will be the smallest, with a, c being somewhere in the middle. So p is less than r, which is less than q. So the question asked us, which of the following statements about the lengths of the paths is true? The answer is p is less than r, which is less than q. Letter b.

equilateral triangle

path lengths

symmetry

relative lengths

P < R < Q