Prove that the center of mass of a thin metal plate in the shape of an equilateral triangle is located at the intersection of the triangle's altitudes by direct calculation and by physical reasoning.

To find the center of mass of a thin metal plate in the shape of an equilateral triangle, we will first determine the coordinates of its vertices. Let's assign vertices A, B, and C with coordinates A(0, 0), B(a, 0), and C(a/2, h), where a is the side length of the triangle and h is the height. Now, we will find the center of mass (x, y) using the formulas: x = (x1 + x2 + x3)/3 y = (y1 + y2 + y3)/3 Plugging the coordinates of A(0, 0), B(a, 0), and C(a/2, h) into the equations, we get: x = (0 + a + a/2)/3 x = a/2 y = (0 + 0 + h)/3 y = h/3 This gives us the center of mass, M(a/2, h/3), which is the intersection point of the altitudes of the equilateral triangle.

To prove this using physical reasoning, consider the triangle as a thin metal plate. We can observe that the plate has three sides, and all sides are equal in length. Since all sides share equal weight, and the shape is symmetrical, the center of mass should lie on the symmetry lines of the triangle. An altitude of an equilateral triangle is a line drawn from a vertex perpendicular to the midpoint of the opposite side. It is also a line of symmetry for the triangle. Thus, the center of mass will lie on the altitude. Since the triangle is equilateral, there are three possible lines of symmetry, which are the altitudes. The center of mass lies on all these altitudes due to the symmetrical shape of the triangle. Because all three altitudes of an equilateral triangle intersect at one point, called the orthocenter, this also tells us that our center of mass will be at that intersection point, which we proved by direct calculation to be M(a/2, h/3).