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

Question

1067 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-16T09:04:25

Distance moved by centroid is

#vec(GG’) ~~ color (green)(2.5#

Coordinates of centroid is obtained using the formula,

#G (x,y) = G ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)#

Given : #A (2,-5), B(-1,2), C(3, -3)#

Centroid #G ((x),(y)) = (((2-1+3)/3),((-5+2-3)/3)) = color(brown)(((4/3),(-2))#

Triangle is dilated about point D(-1,3) by a factor of 2/5

#bar(A’D) = (2/5) bar(AD)#

#a’ - d = (2/5) (a - d) = 0.4 a - 0.4d#

#a’((x),(y)) = 0.4((2),(-5)) + 0.6 ((-1),(2)) = color(blue)(((0.2),(-0.8))#

#b’((x),(y)) = 0.4((-1),(2)) + 0.6 ((-1),(2)) = color(blue)(((1),(2))#

#c’((x),(y)) = 0.4((3),(-3)) + 0.6 ((-1),(2)) = color(blue)(((0.6),(0))#

New centroid #G’((x),(y)) = (((0.2+1+0.6)/3),((-0.8+2+0)/3)) = color(brown)(((0.6),(0.4))#

Using distance formula, we can calculate the distance moved by centroid

#vec(GG’) = sqrt((4/3 - 0.6)^2 + (0.4+2)^2) ~~ color (green)(2.5#