Recall that a median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the side opposite the vertex. All triangles have three medians and these three medians intersect in one point called the centroid, shown below. The centroid partitions each median in a 2:1 ratio.
Find the coordinates of the centroid, given the coordinates of the vertices of the triangle as shown.
Proving Triangle Similarity
Start by drawing the right triangles. Below, the base and height of each triangle has been labeled in green.
Finding Length and Height
Because the triangles are similar, the ratios between pairs of corresponding sides are equal. In particular, you know:
Finding the Coordinates of a Point
Earlier, you were asked to find the coordinates of the centroid, given the coordinates of the vertices of the triangle as shown.
Looking at the picture, these coordinates for the centroid are realistic.
The midpoint of a line segment is the point exactly in the middle of the line segment. In what ratio does a midpoint partition a segment?
1:1, because the segments connecting the midpoint to each endpoint will be the same length.
This is the midpoint formula.
Find the midpoint of each of the following segments defined by the given endpoints.
9. Why are the answers to 7 and 8 different?
To see the Review answers, open this PDF file and look for section 10.6.