A line segment has endpoints at (2 ,4 ) and (5 ,3 ). The line segment is dilated by a factor of 3 around (3 ,8 ). What are the new endpoints and length of the line segment?

1 Answer
Alan P.
Dec 29, 2017

New end points: hat(P): (0,-4) and hat(Q):(12,5)
New segment length: abs(hat(P)hat(Q))=15

Explanation:

Let P be the original point (2,4)
Let Q be the original point (5,3)
and
Let C be the center of dilation, (3,8)

Consider the vector vec(CP)
color(white)("XXX")vec(CP)=(2,4)-(3,8)=(-1,-4)
Dilation by a factor of 3 will scale this vector up by a factor of 3
So P will move to the new location:
color(white)("XXX")hat(P)=C+3vec(CP)
color(white)("XXXX")=(3,8)+3(-1,4)
color(white)("XXXX")=(3-3,8-12)
color(white)("XXXX")=(0,-4)

Similarly
color(white)("XXX")vec(CQ)=(5,3)-(2,4)=(3,-1)
and new location for Q at
color(white)("XXX")hat(Q)=(3,8)+3(3,-1)
color(white)("XXXX")=(12,5)

The length of the new line segment will be (using the Pythagorean Theorem)
color(white)("XXX")abs(hat(P)hat(Q))=sqrt((12-0)^2+(5-(-4))^2)
color(white)("XXX")=sqrt(12^2+9^2)
color(white)("XXX")=sqrt(225)
color(white)("XXX")=15

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