Exercise 43 Page 669

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

We are asked to find the coordinates of the centroid of △ ABC. 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] C & AB A & BC B & CA

Let's find the medians of △ ABC and then we can find their point of intersection.

Median of AB

First, we will find the midpoint of AB by substituting the coordinates of A and B into the Midpoint Formula.

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

SubstitutePoints

Substitute ( - 1,11) & ( - 5,1)

(- 1+( - 5)/2,11+ 1/2)

AddNeg

a+(- b)=a-b

(- 1-5/2,11+1/2)

AddSubTerms

Add and subtract terms

(- 6/2,12/2)

CalcQuot

Calculate quotient

(- 3,6)

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

Median of BC

Using the same process, we can find the midpoint of BC. This time we will substitute the coordinates of B and C into the Midpoint Formula.

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

▼

Substitute coordinates and evaluate

SubstitutePoints

Substitute ( - 5,1) & ( - 9,6)

(- 5+( - 9)/2,1+ 6/2)

AddNeg

a+(- b)=a-b

(- 5-9/2,1+6/2)

AddSubTerms

Add and subtract terms

(- 14/2,7/2)

CalcQuot

Calculate quotient

(- 7,3.5)

Now, let's plot the midpoint of BC at (- 7,3.5) and name it K. If we draw a segment from K to vertex A, we will have the median of BC.

Median of CA

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

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

▼

Substitute coordinates and evaluate

SubstitutePoints

Substitute ( - 1,11) & ( - 9,6)

(- 1+( - 9)/2,11+ 6/2)

AddNeg

a+(- b)=a-b

(- 1-9/2,11+6/2)

AddSubTerms

Add and subtract terms

(- 10/2,17/2)

CalcQuot

Calculate quotient

(- 5,8.5)

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

Coordinates of the Midpoint

As we can see on our diagram, the medians intersect at point (- 5,6). These are the coordinates of the centroid of △ ABC.