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

1 Answer
Alan P.
Dec 30, 2017

New endpoints: Dilation of (2,1) rarr (-1,1)(2,1)→(−1,1)
color(white)("New endpoints:")New endpoints:Dilation of (4,2) rarr (7,5)(4,2)→(7,5)
New line segment length: 4sqrt(5)" units"4√5 units

Explanation:

If the center of dilation is C=(3,1)C=(3,1)
and original endpoints are
{: (P=(2,1),color(white)("xxxxx"),Q=(4,2)), (vec(CP)=(2-3,1-1),,vec(CQ)=(4-3,2-1)), (color(white)(vec(CP))=(-1,0),,color(white)(vec(CQ))=(1,1)), ("dilated vector:",,"dilated vector:"), (4vec(CP)=(-4,0),,4vec(CQ)=(4,4)), (hat(P)=C+4vec(CP),,hat(Q)=C+4vec(CQ)), (color(white)(hat(P))=(-1,1),,color(white)(hat(Q))=(7,5)) :}

The length of the dilated line segment is
abs(hatPhatQ)=sqrt((7-(-1))^2+(5-1)^2)=sqrt(80)=4sqrt(5)

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