Exercise 44 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 & ZX

Let'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.

(x_1+x_2/2,y_1+y_2/2)

SubstitutePoints

Substitute ( 4,0) & ( - 2,4)

(4+( - 2)/2,0+ 4/2)

AddNeg

a+(- b)=a-b

(4-2/2,0+4/2)

AddSubTerms

Add and subtract terms

(2/2,4/2)

CalcQuot

Calculate quotient

(1,2)

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.

(x_1+x_2/2,y_1+y_2/2)

▼

Substitute coordinates and evaluate

SubstitutePoints

Substitute ( - 2,4) & ( 0,6)

(- 2+ 0/2,4+ 6/2)

AddTerms

Add terms

(- 2/2,10/2)

CalcQuot

Calculate quotient

(- 1,5)

Now, let's plot the midpoint of YZ at (- 1,5) 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.

(x_1+x_2/2,y_1+y_2/2)

▼

Substitute coordinates and evaluate

SubstitutePoints

Substitute ( 4,0) & ( 0,6)

(4+ 0/2,0+ 6/2)

AddTerms

Add terms

(4/2,6/2)

CalcQuot

Calculate quotient

(2,3)

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 ( 23, 103). These are the coordinates of the centroid of △ XYZ.