Exercise 41 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 ( 2,2) & ( 7,8)

(2+ 7/2,2+ 8/2)

AddTerms

Add terms

(9/2,10/2)

CalcQuot

Calculate quotient

(4.5,5)

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 ( 7,8) & ( 12,2)

(7+ 12/2,8+ 2/2)

AddTerms

Add terms

(19/2,10/2)

CalcQuot

Calculate quotient

(9.5,5)

Now, let's plot the midpoint of BC at (9.5,5) 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 ( 2,2) & ( 12,2)

(2+ 12/2,2+ 2/2)

AddTerms

Add terms

(14/2,4/2)

CalcQuot

Calculate quotient

(7,2)

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 (7,4). These are the coordinates of the centroid of △ ABC.