Question

A triangle has corners at `(8 ,3 )`, `(4 ,-6 )`, and `(-2 ,-4 )`. If the triangle is dilated by a factor of `5` about point #(1 ,-3 ), how far will its centroid move?

Answer 1

Distance moved by centroid `vec(GG’) = color (green)(15.6205`

Explanation:

Given : `A (8,3), B (4,-6), C (-2,-4)` Dilated about `D(1,-3)` by a factor of 5.

To find the distance moved by centroid.

`vec(A’D) = 5 * vec(AD)`

`a’ - d = 5a - 5d` or `a’ = 5a - 4d`

`a’((x),(y)) = 5((8),(3)) - 4 ((1),(-3)) = ((40),(15)) - ((4),(-12)) = ((36),(27))`

Similarly,

`b’((x),(y)) = 5((4),(-6)) - 4 ((1),(-3)) = ((20),(-30)) - ((4),(-12)) = ((16),(18))`

`b’((x),(y)) = 5((-2),(-4)) - 4 ((1),(-3)) = ((-10),(-20)) - ((4),(-12)) = ((-6),(-8))`

Original centroid `g((x),(y)) = (((8+4-2)/3),((3-6-4)/3)) = ((10/3),(-7/3))`

New centroid after dilation

`g’((x),(y)) = (((36 + 16 -6)/3),((27+18-8)/3)) = ((46/3),(37/3))`

Distance moved by centroid

`vec(GG’) = sqrt((10/3-46/3)^2+ (-7/3 + 37/3)^2) = sqrt244 = color (green)(15.6205`