Exercise 42 Page 669

Let's start by plotting the given points on a coordinate plane and drawing triangle △ XYZ.

We are asked to find the coordinates of the centroid of △ XYZ. This is the point of concurrency of the triangle's medians. A median of a triangle is a segment with one endpoint being a vertex and the other endpoint being the midpoint of the opposite side. cc Vertex & Opposite Side [0.8em] Z & XY X & YZ Y & ZXLet's find the medians of △ XYZ and then we can find their point of intersection.

Median of XY

First, we will find the midpoint of XY by substituting the coordinates of X and Y into the Midpoint Formula.

Now that we know the coordinates of the midpoint of XY, we can plot this point on our diagram. Let's name this point N. Drawing a segment from Z to N, we will get the median of XY.

Median of YZ

Using the same process, we can find the midpoint of YZ. This time we will substitute the coordinates of Y and Z into the Midpoint Formula.

Now, let's plot the midpoint of YZ at (- 3,- 9.5) and name it K. If we draw a segment from K to vertex X, we will have the median of YZ.

Median of ZX

One last time, we will follow the same procedure — this time using the coordinates of X and Z.

Let's name this midpoint L and add it to the diagram. The segment that connects Y and L is a median of XZ.

Coordinates of the Midpoint

As we can see on our diagram, the medians intersect at point (- 3,- 7). These are the coordinates of the centroid of △ XYZ.