A triangle as corners at #(5 ,2 )#, #(2 ,4 )#, and #(3 ,8 )#. If the triangle is dilated by a factor of #3 # about #(2 ,3 ), how far will its centroid move?

Question

A triangle as corners at #(5  ,2    )#, #(2   ,4   )#, and #(3  ,8    )#.  If the triangle is dilated by a factor of #3 # about #(2  ,3     ), how far will its centroid move?

Sadie Allen · Accepted Answer

centroid G has moved by a distance of #~~ color(green)(4.27# units rounded to 2 decimals

Given : A(5,2), B(2,4), C = (3,8), D(2,3)#

Dilated around D and by a factor of 3

To find the distance moved by the centroid #vec(GG')#

Coordinates of #G = (5 + 2 + 3)/3, (2+4+6)/3 = color(purple)((10/3, 14/3)#

#vec(A'D) = 3 * (vec(AD))#

#a' = 3a - 2d = 3((5),(2)) - 2((2),(3)) =color(blue)( ((11),(0))#

Similarly, #b' = 3b - 2d = 3((2),(4)) - 2((2),(3)) = color(blue)(((2),(6))#

and #c' = 3c - 2d = 3((3),(8)) - 2((2),(3)) = color(blue)(((5),(18))#

Coordinates of #G' = (11 + 2 + 5)/3, (0 + 6 + 18)/3 =color(purple)( (6,8)#

Using distance formula to find the distance between G & G',

#vec(GG') = sqrt((6-(10/3))^2 + (8-(14/3))^2) ~~ color(green)(4.27# rounded to 2 decimals

Wyatt Anderson · Answer

undefined To find how far the centroid of the triangle will move after dilation, you need to understand the properties of dilation and the centroid of a triangle.

The centroid of a triangle is the point of concurrency of its medians, which are the lines joining each vertex to the midpoint of the opposite side. Dilation changes the size of the triangle but does not change its shape or orientation, so the centroid remains on the same line segment, known as the centroid's median.

To find the new centroid after dilation, you first need to find the coordinates of the original centroid, which can be calculated as the average of the coordinates of the triangle's vertices.

After dilation, the new coordinates of the triangle's vertices can be found by multiplying the distance from the center of dilation to each vertex by the dilation factor and adding the result to the original coordinates of the center of dilation.

Then, you can find the new centroid of the dilated triangle by calculating the average of the coordinates of the new vertices.

Finally, you can determine how far the centroid has moved by finding the distance between the original and new centroid coordinates.