A line segment has endpoints at $(9, 3)$ $(9 ,3 )$ and $(5, 4)$ $(5 ,4 )$ . The line segment is dilated by a factor of $3$ $3$ around $(4, 6)$ $(4 ,6 )$ . What are the new endpoints and length of the line segment?

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Answer 1 · 2018-04-26T14:16:21

Endpoints $(19, - 3) and (7, 0)$ $(19,-3) and (7,0)$ and length $3 \sqrt{17} .$ $3sqrt{17}.$

Let $D = (4.6)$ $D=(4.6)$ be the dilation point, $r$ $r$ be the dilation factor, and $P$ $P$ be the point being dilated.

I always think of translating the dilation point to the origin, dilating, then translating back. So $P'$ , the image of $P$ , is

$P' = r (P-D) + D = rP + (1-r)D$

That's the standard parametric form for a line from $D$
to $P$ ,

$l(t)=(1-t)D+tP$ .

We have $l(0)=D$ , $l(1)=P$ and our image $P'=l(r)$ . That makes sense; dilation sends points off along a ray from the dilation point.

We can precompute

$(1-r)D = (1-3)D=-2(4,6)=(-8,-12).$

The image of $(9,3)$ is $3(9,3)+(-8,-12)=(19,-3)$

The image of $(5,4)$ is $3(5,4)+(-8,-12)=(7,0)$

The length will be three times the original length,

$l = 3\sqrt{ (9-5)^2+(3-4)^2} = 3sqrt{17}$

We can check that.

$sqrt{ (19-7)^2 + (-3)^2}= sqrt{144+9}=sqrt{153}=3 sqrt{17} quad sqrt$

A line segment has endpoints at (9 ,3 )(9,3)(9 ,3 ) and (5 ,4 )(5,4)(5 ,4 ). The line segment is dilated by a factor of 3 33 around (4 ,6 )(4,6)(4 ,6 ). What are the new endpoints and length of the line segment?