A line segment has endpoints at $(3 ,8 )$ and $(5 , 9)$ . The line segment is dilated by a factor of $3$ around $(2 , 3)$ . What are the new endpoints and length of the line segment?

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Answer 1 · 2018-05-23T19:41:56

$color(blue)(A'=(5,18)$
$color(blue)(B'=(11,21)$
$color(blue)(d=3sqrt(5))$

We can preform a dilation using vectors.

Let $A=(3,8)$ , $B=(5,9)$ , $D=(2,3)$ and $O$ be the origin.

$vec(OD)=((2),(3))$

$vec(DA)=((1),(5))$

$vec(DB)=((3),(6))$

Dilating A:

$vec(OA)=vec(OD)+3vec(DA)=((2),(3))+3((1),(5))=((5),(18))$

Dilating B:

$vec(OB)=vec(OD)+3vec(DB)=((2),(3))+3((3),(6))=((11),(21))$

New endpoints:

$A'=(5,18)$ and $B'=(11,21)$

Length of line segment is found using the distance formula:

$d=sqrt((11-5)^2+(21-18)^2)=sqrt(45)=color(blue)(3sqrt(5))$

A line segment has endpoints at (3 ,8 )(3 ,8 ) and (5 , 9)(5 , 9). The line segment is dilated by a factor of 3 3 around (2 , 3)(2 , 3). What are the new endpoints and length of the line segment?