Exercise 11 Page 434

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

To find the location of the centroid, we need to recall two definitions.

1. The centroid describes the point of concurrency of the triangle's medians.
2. A medians of a triangle is a segment with one endpoint being a vertex and the other endpoint being the midpoint of the opposite side.

   Let's draw the medians of our triangle.

   

   To find the coordinates of P, we will first find the coordinates of the midpoint D of RT. Then, we will use the Centroid Theorem.

   Midpoint of RT

   To find the coordinates of D, we will substitute the coordinates of R and T into the Midpoint Formula.
   (x_1+x_2/2,y_1+y_2/2)

   SubstitutePoints

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

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

   AddTerms

   Add terms

   (- 4/2,8/2)

   CalcQuot

   Calculate quotient

   (- 2,4)
   The coordinates of D are (- 2,4). We can plot this point on our diagram and focus the median of RT.
   

   Using the Centroid Theorem

   By the Centroid Theorem, we know that the centroid is two thirds of the distance from each vertex to the midpoint of the opposite side. Thus, we have PS= 23DS.

   
   We see that DS is a vertical line. The distance from D( - 2, 4) to S( - 2, - 2) is 6 units. 4 - ( - 2) = 6 Now that we know DS, we can find PS.
   PS = 2/3 DS

   Substitute

   DS= 6

   PS=2/3( 6)

   MoveRightFacToNum

   a/c* b = a* b/c

   PS=12/3

   CalcQuot

   Calculate quotient

   PS=4
   This means that the centroid is 4 units up from S. Therefore, adding 4 to the y-coordinate of S( - 2, - 2) will give us the coordinates of the centroid. ( - 2, - 2+ 4 ) ⇔ (- 2,2)