A line segment has endpoints at (3 ,2 )(3,2) and (7 , 5)(7,5). The line segment is dilated by a factor of 4 4 around (2 , 3)(2,3). What are the new endpoints and length of the line segment?

1 Answer
Jim G.
Apr 21, 2017

(6,-1),(22,11),20" units"(6,−1),(22,11),20 units

Explanation:

let A=(3,2),B=(7,5)" and " C=(2,3)A=(3,2),B=(7,5) and C=(2,3)

"and " A',B'" be the image of A and B under the dilatation"

vec(CA)=ula-ulc=((3),(2))-((2),(3))=((1),(-1))

rArrvec(CA')=4((1),(-1))=((4),(-4))

rArrA'=(2+4,3-4)=(6,-1)color(red)(larr)

vec(CB)=ulb-ulc=((7),(5))-((2),(3))=((5),(2))

rArrvec(CB')=4((5),(2))=((20),(8))

rArrB'=(2+20,3+8)=(22,11)color(red)(larr)

To calculate length use the color(blue)"distance formula"

color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))

"let " (x_1,y_1)=(6,-1)" and " (x_2,y_2)=(22,11)

d_(A'B')=sqrt((22-6)^2+(11+1)^2)

color(white)(d_(A'B'))=sqrt(256+144)=sqrt400

rArrd_(A'B')=20" units"

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