A triangle has corners at #(7 ,3 )#, #(9 ,4 )#, and #(5 ,2 )#. If the triangle is dilated by a factor of #2 # about point #(6 ,1 ), how far will its centroid move?

Question

A triangle has corners at #(7  ,3    )#, #(9   ,4   )#, and #(5  ,2    )#.  If the triangle is dilated by a factor of #2 # about point #(6  ,1     ), how far will its centroid move?

Wyatt Anderson · Accepted Answer

#color(purple)(vec(GG') = sqrt((22/3-7)^2 + (5-3)) ~~ 2.0276 " units"# #A(7,3), B(9,4), C(5,2), " about point " D (6,1), " dilation factor "2# Centroid #G(x,y) = ((x_a + x_b + x_c) /3, (y_a + y_b + y_c)/3)# #G(x,y) = ((7+9+5)/3, (3+4+2)/3) = (7, 3)# #A'((x),(y)) = 2a - d = 2*((7),(3)) - ((6),(1)) = ((8),(5))# #B'((x),(y)) = 2b - d = 2*((8),(4)) - ((6),(1)) = ((10),(7))# #C'((x),(y)) = 2c - d = 2*((5),(2)) - ((6),(1) = ((4),(3))# #"New Centroid " G'(x,y) = ((8+10+4)/3,(5+7+3)/3) = (22/3,5)# #color(purple)("Distance moved by centroid " # #color(purple)(vec(GG') = sqrt((22/3-7)^2 + (5-3)) ~~ 2.0276 " units"#

Noah Atkinson · Answer

undefined To find how far the centroid of the triangle moves when dilated by a factor of 2 about point (6, 1), follow these steps:

1. Find the coordinates of the centroid of the original triangle.
2. Dilate the original triangle by a factor of 2 about point (6, 1) to find the new coordinates of the vertices.
3. Find the coordinates of the centroid of the dilated triangle.
4. Calculate the distance between the centroids of the original and dilated triangles.

First, find the coordinates of the centroid of the original triangle:

Centroid coordinates = ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3)

Using the given coordinates:
Centroid coordinates = ((7 + 9 + 5) / 3, (3 + 4 + 2) / 3)
                      = (21 / 3, 9 / 3)
                      = (7, 3)

Now, dilate the original triangle by a factor of 2 about point (6, 1):

For each vertex (x, y), the new coordinates are:
New_x = 2 * (x - 6) + 6
New_y = 2 * (y - 1) + 1

Using the given vertices:
Vertex 1: (7, 3)
New_x1 = 2 * (7 - 6) + 6 = 8
New_y1 = 2 * (3 - 1) + 1 = 5

Vertex 2: (9, 4)
New_x2 = 2 * (9 - 6) + 6 = 12
New_y2 = 2 * (4 - 1) + 1 = 7

Vertex 3: (5, 2)
New_x3 = 2 * (5 - 6) + 6 = 4
New_y3 = 2 * (2 - 1) + 1 = 3

Now, find the coordinates of the centroid of the dilated triangle:

Centroid coordinates = ((New_x1 + New_x2 + New_x3) / 3, (New_y1 + New_y2 + New_y3) / 3)

Centroid coordinates = ((8 + 12 + 4) / 3, (5 + 7 + 3) / 3)
                      = (24 / 3, 15 / 3)
                      = (8, 5)

Finally, calculate the distance between the centroids of the original and dilated triangles:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)
         = √((8 - 7)^2 + (5 - 3)^2)
         = √(1^2 + 2^2)
         = √(1 + 4)
         = √5

So, the centroid of the original triangle moves √5 units when the triangle is dilated by a factor of 2 about point (6, 1).