Exercise 13 Page 340

To find the place for the string, we need to find the balancing point. In a triangle, the balancing point is a centroid of the triangle. Hence, we should locate the centroid of the triangle. First, using the given coordinates, we will draw the triangle on a coordinate plane.

Let's recall that a centroid of a triangle is point of intersection of the triangle medians. Thus, to locate the centroid, we can draw the medians of our triangle and find their point of concurrency. The other way to find the centroid is to use the Centroid Theorem. Let's review what it states. The medians of a triangle inersect at a point called the centroid that is two thirds of the distance from each vertex to the midpoint of the opposite side.First, we can draw a median from the vertex with the coordinates (6,4). To do this, we need to know the coordinates of the midpoint of the opposite side. Let's find them by substituting (0,8) and (3,0) into Midpoint Formula.

Now we can plot M(1.5,4) on the coordinate plane and draw the median. Let's also name the vertices of the triangle for the following explanation to be easier.

According to the above theorem, the centroid of △ NLK is two thirds of the distance from L to M. If we call the centroid C, the following equality is true. LC= 23LM To find LM, let's examine the diagram. We can see that L(6, 4) and M(1.5, 4) have the same y-coordinate, 4. Hence, to find the measure of LM, we need to calculate the difference between their x-coordinates. LM=6-1.5=4.5 Now we can substitute 4.5 for LM into the above equation and calculate LC.

The measure of LC is 3. Since C is located to the left of L, we can find the x-coordinate of C by subtracting 3 from the x-coordinate of L(6, 4). C(6-3, 4) ⇒ C(3,4) Therefore, the coordinates of the centroid of the triangle are (3,4).