Coordinates of Midpoints and the Centroid

This is an investigation to find the coordinates of the midpoints of a triangle's three sides and the coordinates of the centroid. Look at the questions below, and drag the triangle's vertices into the correct positions to help you answer the questions.
1. For A(1, 7), B(1, 1), C(4, 7),
  • classify the triangle;
  • what are the coordinates of the three midpoints?
  • what are the coordinates of the centroid?
  • when you look at the coordinates of the midpoint of AB, can you see a way you could have calculated these coordinates using the coordinates of A and B?
  • when you look at the coordinates of the midpoint of B and C, can you see a way to calculate these coordinates using the coordinates of B and C?
  • when you look at the coordinates of the centroid, can you see a way to calculate these coordinates from the coordinates of A, B, and C? (Hint: how could you get the x-value of the centroid from the x-values of the three vertices?)
2. For A(1, 0), B(3, 7), C(8, 5),
  • classify the triangle;
  • show how to use the coordinates of A, B, and C to calculate the coordinates of the three midpoints;
  • show how to use the coordinates of A, B, and C to calculate the coordinates of the centroid.
3. Using the same triangle as in #2, consider the median coming from point B.
  • What is the distance from B to the centroid of the triangle? Express this distance to the nearest hundredth of a unit.
  • What is the distance from the centroid to the midpoint of side AC? Express this distance to the nearest hundredth of a unit.
  • What is the relationship between these two distances?