A triangle has corners at $(1 ,9 )$ , $(-6 ,-1 )$ , and $(1 ,-2 )$ . If the triangle is dilated by a factor of $2$ about point #(-5 ,-2 ), how far will its centroid move?

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Answer 1 · 2018-05-25T10:02:36

$color(blue)((sqrt(265))/3" units")$

If all vertices of a triangle are dilated by a factor $bba$ then the centroid will also be dilated by a factor $bba$

Let $A$ be the triangle in its original position.

Let $A'$ be the dilated triangle.

Centroid of $A$

The coordinates of the centroid can be found by taking the arithmetic mean of the x coordinates and the y coordinates.

$((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3)$

$((1-6+1)/3,(9-1-2)/3)=>(-4/3,2)$

Using vectors:

Position vector of centre of dilation:

$vec(OD)=((-5),(-2))$

Let $C$ be the centroid of $A'$ .

Vector from D to C:

$vec(DC)=((11/3),(4))$

Scale factor = 2:

Position vector of new centroid $C'$

$vec(OD)+3vec(DC)=((-5),(-2))+2((11/3),(4))=((7/3),(6))$

Distance the centroid has moved can be found using the distance formula:

$d=sqrt((-4/3-7/3)^2+(2-6)^2)=(sqrt(265))/3$ units

A triangle has corners at (1 ,9 )(1 ,9 ), (-6 ,-1 )(-6 ,-1 ), and (1 ,-2 )(1 ,-2 ). If the triangle is dilated by a factor of 2 2 about point #(-5 ,-2 ), how far will its centroid move?