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 & 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 ( - 3,- 2) & ( 1,- 12)

(- 3+ 1/2,- 2+( - 12)/2)

AddNeg

a+(- b)=a-b

(- 3+1/2,- 2-12/2)

AddSubTerms

Add and subtract terms

(- 2/2,- 14/2)

CalcQuot

Calculate quotient

(- 1,- 7)

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 ( 1,- 12) & ( - 7,- 7)

(1+( - 7)/2,- 12+( - 7)/2)

AddNeg

a+(- b)=a-b

(1-7/2,- 12-7/2)

SubTerms

Subtract terms

(- 6/2,- 19/2)

CalcQuot

Calculate quotient

(- 3,- 9.5)

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.

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

▼

Substitute coordinates and evaluate

SubstitutePoints

Substitute ( - 3,- 2) & ( - 7,- 7)

(- 3+( - 7)/2,- 2+( - 7)/2)

AddNeg

a+(- b)=a-b

(- 3-7/2,- 2-7/2)

SubTerms

Subtract terms

(- 10/2,- 9/2)

CalcQuot

Calculate quotient

(- 5,- 4.5)

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.