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

1 Answer
Leland Adriano Alejandro
Jun 22, 2016

distance d=5.3851648" "units

Explanation:

Compute for the centroid (x_c, y_c)

x_c=(x_1+x_2+x_3)/3=(1+9+2)/3=4
y_c=(y_1+y_2+y_3)/3=(3+4+2)/3=3

Centroid (x_c, y_c)=(4, 3)

Factor of 2 about the point (9, 5)

Let (x_c', y_c') be the new centroid

Solve for x_c'

(x_c'-9)/(x_c-9)=2/1

(x_c'-9)/(4-9)=2/1

(x_c'-9)/(-5)=2

x_c'=-10+9

x_c'=-1

Solve for y_c'

(y_c'-5)/(y_c-5)=2/1

(y_c'-5)/(3-5)=2/1

(y_c'-5)/(-2)=2

y_c'=-4+5

y_c'=1

The new centroid (x_c', y_c')=(-1, 1)

Solve for the distance between the two centroids

d=sqrt((x_c-x_c')^2+(y_c-y_c')^2)

d=sqrt((4--1)^2+(3-1)^2)

d=sqrt((5)^2+(2)^2)

d=sqrt(25+4)

d=sqrt(29)

d=5.3851648" "units

KIndly see the drawings of the old and new triangles
enter image source here

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

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