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

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

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Answer 1 · 2017-12-29T14:52:47

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

Explanation:

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

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

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

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

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

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Explanation:

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1 Answer

Explanation:

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