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There is a triangle with 6 points, (0,0), (a/2,0), (a,0), (a+b/2, c/2), (b,c), and (b/2, c/2)

One way to prove that the medians of a triangle all meet at one point is by using arbitrary coordinates (0,0), (a,0), and (b,c) to represent the three vertices of the triangle. Drag and drop the missing pieces of each step to complete the proof that all three medians intersect at point P.

Step 1: Draw the three medians of the triangle. Draw each median so that it starts at a ____ and ends at the ____ 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 (b/2, c/2). The midpoint of (0, 0) and (a, 0) is (a/2, 0). The midpoint of (a, 0) and (b, c) is (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 _____

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).

1/3(0,0)+2/3(a+b/2, c/2)=(0,0)+(a+b/3, c/3)=(a+b/3, c/3)

1 answer

Let's complete the proof by filling in the missing pieces for each step.

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 (b/2, c/2). The midpoint of (0, 0) and (a, 0) is (a/2, 0). The midpoint of (a, 0) and (b, c) is (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 for finding the centroid of a triangle, which is given by P = (1/3)(x1, y1) + (1/3)(x2, y2) + (1/3)(x3, y3) where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices of the triangle.

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). 1/3(0,0)+2/3(a+b/2, c/2)=(0,0)+(a+b/3, c/3)=(a+b/3, c/3).

By substituting the values for the other medians, you'll find that all calculations for point P will yield the same result, verifying that all three medians intersect at the centroid of the triangle.