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

1 Answer

The centroid will move by about d=4/3sqrt233=20.35245" "units

Explanation:

We have a triangle with vertices or corners at the points A(-6, 3) and B(3, -2) and C(5, 4).

Let F(x_f, y_f)=F(-2, 6)" "the fixed point

Compute the centroid O(x_g, y_g) of this triangle, we have

x_g=(x_a+x_b+x_c)/3=(-6+3+5)/3=2/3
y_g=(y_a+y_b+y_c)/3=(3+(-2)+4)/3=5/3

Centroid O(x_g, y_g)=O(2/3, 5/3)

Compute the centroid of the bigger triangle (scale factor =5)

Let O'(x_g', y_g')=the centroid of the bigger triangle

the working equation:

(FO')/(FO)=5

solve for x_g':

(x_g'--2)/(2/3--2)=5

(x_g'+2)=5*8/3
x_g'=40/3-2
x_g'=34/3

solve for y_g'

(y_g'-6)/(5/3-6)=5

y_g'=6+5(-13/3)=(18-65)/3

y_g'=-47/3

Compute now the distance from centroid O(2/3, 5/3) to new centroid O'(34/3, -47/3).

d=sqrt((x_g-x_g')^2+(y_g-y_g')^2)

d=sqrt((2/3-34/3')^2+(5/3--47/3)^2)

d=sqrt((-32/3)^2+(52/3)^2)

d=sqrt(((-4*8)/3)^2+((4*13)/3)^2)

d=4/3*sqrt(64+169)

d=4/3*sqrt(233)=20.35245

God bless....I hope the explanation is useful..