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

Question

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

Sophia Alfred · Accepted Answer

The distance is #=5.83#

Let #ABC# be the triangle

#A=(5.3)#

#B=(1,4)#

#C=(3.5)#

The centroid of triangle #ABC# is

#C_c=((5+1+3)/3,(3+4+5)/3)=(3,4)#

Let #A'B'C'# be the triangle after the dilatation

The center of dilatation is #D=(2,2)#

#vec(DA')=3vec(DA)=3*<3,1> = <9,3>#

#A'=(9+2,3+2)=(11,5)#

#vec(DB')=3vec(DB)=3*<-1,2> = <-3,6>#

#B'=(-3+2,6+2)=(-1,11)#

#vec(DC')=3vec(Dc)=3*<2,3> = <6,9>#

#C'=(6+2,9+2)=(8,11)#

The centroid #C_c'# of triangle #A'B'C'# is

#C_c'=((11-1+8)/3,(5+11+11)/3)=(6,9)#

The distance between the 2 centroids is

#C_cC_c'=sqrt((6-3)^2+(9-4)^2)#

#=sqrt(9+25)=sqrt34=5.83#

Noah Atkinson · Answer

undefined To find how far the centroid of the triangle moves when it is dilated by a factor of 3 about the point (2, 2), we need to first calculate the centroid of the original triangle and then find its new coordinates after dilation.

The centroid of a triangle is the point where the three medians intersect. The medians of a triangle are the lines joining each vertex to the midpoint of the opposite side.

For the original triangle with vertices (5, 3), (1, 4), and (3, 5), the midpoints of the sides are:

Midpoint of (5, 3) and (1, 4):
$$ \left( \frac{5 + 1}{2}, \frac{3 + 4}{2} \right) = \left( 3, \frac{7}{2} \right) $$

Midpoint of (1, 4) and (3, 5):
$$ \left( \frac{1 + 3}{2}, \frac{4 + 5}{2} \right) = \left( 2, \frac{9}{2} \right) $$

Midpoint of (3, 5) and (5, 3):
$$ \left( \frac{3 + 5}{2}, \frac{5 + 3}{2} \right) = \left( 4, 4 \right) $$

Now, we find the centroid, which is the average of the vertices:

Centroid: 
$$ \left( \frac{5 + 1 + 3}{3}, \frac{3 + 4 + 5}{3} \right) = \left( \frac{9}{3}, \frac{12}{3} \right) = \left( 3, 4 \right) $$

After dilation by a factor of 3 about the point (2, 2), the new coordinates of the centroid will be three times the distance from the center of dilation to the original centroid, added to the coordinates of the center of dilation:

New centroid coordinates:
$$ x_{new} = 3 \times (2 - 3) + 2 = -1 + 2 = 1 $$
$$ y_{new} = 3 \times (2 - 4) + 2 = -6 + 2 = -4 $$

So, the centroid moves to the point (1, -4). Therefore, the centroid moves a distance of 5 units horizontally and 8 units vertically.