A line segment has endpoints at $(9, 7)$ $(9 ,7 )$ and $(1, 2)$ $(1 ,2 )$ . The line segment is dilated by a factor of $4$ $4$ around $(3, 3)$ $(3 ,3 )$ . What are the new endpoints and length of the line segment?

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Answer 1 · 2018-05-17T14:25:03

$color(blue)((27,19) \ \ "and" \ (-5,-1)$

One method to perform these is by using vectors.

Let:

$bba$ be position vector $((9),(7))$

$bb(b)$ be position vector $((1),(2))$

$bbd$ be position vector $((3),(3))$

$vec(bb(da))=((6),(4))$

Dilations by a factor 4:

$4vec(bb(da))=4((6),(4))=((24),(16))$

Position vector of the image of $bba$ :

We now just add:

$a'=bbd+4vec(bb(da))=((3),(3))+4((6),(4))=((27),(19))$

$:.$

$(9,7)->(27,19)$

We do the same for point b:

$vec(bb(db))=((-2),(-1))$

Position vector of the image of $bb(b)$

$bb(b)+4vec(bb(db))=((3),(3))+4((-2),(-1))=((-5),(-1))$

$(1,2)->(-5,-1)$

So new endpoints are:

$(27,19),(-5,-1)$

PLOT:

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A line segment has endpoints at (9 ,7 )(9,7)(9 ,7 ) and (1 ,2 )(1,2)(1 ,2 ). The line segment is dilated by a factor of 4 44 around (3 ,3 )(3,3)(3 ,3 ). What are the new endpoints and length of the line segment?