Step 1: Draw the three medians of the triangle. Draw each median so that it starts at a Response area and ends at the Response area of the opposite side. Step 2: Find the coordinates of each midpoint by taking the sum of the x-coordinates and dividing by 2 and taking the sum of the y-coordinates and dividing by 2. The midpoint between (0, 0) and (b, c) is (b2, c2) . The midpoint of (0, 0) and (a, 0) is (a2, 0) . The midpoint of (a, 0) and (b, c) is (a+b2, c2) . Step 3: Find the coordinates of P, the point where the medians appear to intersect, on each median. Use the formula Response area. Step 4: Find the coordinates of P on the median that starts at vertex (0, 0) and ends at midpoint (a+b2, c2) . 13(0, 0)+23(a+b2, c2)=(0, 0) + (a+b3, c3)=(a+b3, c3) Find the coordinates of P on the median that starts at vertex Response area and ends at midpoint (b2, c2) . 13(a, 0)+23(b2, c2)=(a3, 0)+(b3,c3)=(a+b3, c3) Find the coordinates of P on the median that starts at vertex (b, c) and ends at midpoint (a2, 0) . 13(b, c)+23(a2, 0)=(b3,c3)+(a3,0)=(a+b3,c3) Step 5: The coordinates of P on each median are Response area, which proves that the three medians of this generic triangle all intersect at the same point.

Question

Step 1: Draw the three medians of the triangle. Draw each median so that it starts at a Response area and ends at the Response area of the opposite side. Step 2: Find the coordinates of each midpoint by taking the sum of the x-coordinates and dividing by 2 and taking the sum of the y-coordinates and dividing by 2. The midpoint between (0, 0) and (b, c) is (b2, c2) . The midpoint of (0, 0) and (a, 0) is (a2, 0) . The midpoint of (a, 0) and (b, c) is (a+b2, c2) . Step 3: Find the coordinates of P, the point where the medians appear to intersect, on each median. Use the formula Response area. Step 4: Find the coordinates of P on the median that starts at vertex (0, 0) and ends at midpoint (a+b2, c2) . 13(0, 0)+23(a+b2, c2)=(0, 0) + (a+b3, c3)=(a+b3, c3) Find the coordinates of P on the median that starts at vertex Response area and ends at midpoint (b2, c2) . 13(a, 0)+23(b2, c2)=(a3, 0)+(b3,c3)=(a+b3, c3) Find the coordinates of P on the median that starts at vertex (b, c) and ends at midpoint (a2, 0) . 13(b, c)+23(a2, 0)=(b3,c3)+(a3,0)=(a+b3,c3) Step 5: The coordinates of P on each median are Response area, which proves that the three medians of this generic triangle all intersect at the same point.

Answer 1

Step 1: Draw the three medians of the triangle. Draw each median so that it starts at a vertex and ends at the midpoint of the opposite side.

Step 2: Find the coordinates of each midpoint by taking the sum of the x-coordinates and dividing by 2, and taking the sum of the y-coordinates and dividing by 2. The midpoint between (0, 0) and (b, c) is ((0+b)/2, (0+c)/2) = (b/2, c/2). The midpoint of (0, 0) and (a, 0) is ((0+a)/2, (0+0)/2) = (a/2, 0). The midpoint of (a, 0) and (b, c) is ((a+b)/2, (0+c)/2) = (a+b/2, c/2).

Step 3: Find the coordinates of P, the point where the medians appear to intersect, on each median. Use the formula P = (1/3)(A) + (2/3)(B), where A and B are the endpoints of the median.

Step 4: Find the coordinates of P on the median that starts at vertex (0, 0) and ends at midpoint (a+b/2, c/2). P = (1/3)(0, 0) + (2/3)(a+b/2, c/2) = (2(a+b)/6, 2c/6) = ((a+b)/3, c/3).

Step 5: Find the coordinates of P on the median that starts at vertex (a, 0) and ends at midpoint (b/2, c/2). P = (1/3)(a, 0) + (2/3)(b/2, c/2) = (a/3 + b/3, 0 + c/3) = ((a+b)/3, c/3).

Step 6: Find the coordinates of P on the median that starts at vertex (b, c) and ends at midpoint (a/2, 0). P = (1/3)(b, c) + (2/3)(a/2, 0) = (b/3 + a/3, c/3 + 0) = ((a+b)/3, c/3).

Step 7: The coordinates of P on each median are ((a+b)/3, c/3), which proves that the three medians of this generic triangle all intersect at the same point.