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

1 Answer
Apr 11, 2018

Please read the explanation.

Explanation:

" "
Given:

enter image source here

Scale Factor for dilation is 3.

Useful observations involving Dilation:

Isometry refers to a linear transformation which preserves the length.

Dilation is NOT an isometry. It creates similar figures only.

Here bar (AB) is the pre-image and after dilation, bar (A'B') is called the image.

The absolute value of the scale factor (k),

with the constraint 0 < k < 1,

reduces the line segment bar (AB),

enlarges if otherwise.

Each point on the line segment bar (AB) will get 3 times as far from the Center of Dilation, (4,3) since the scale factor is 3.

Dilation preserves the angle of measure.

enter image source here

Note that the pre-image and the image are parallel.

Observe that the points (center of dilation color(red)C , A and A') collinear.

And, the points (C, B and B') are also collinear.

bar (AB) |\| bar (A'B'), since we have congruent corresponding angles.

Also, from C(4,3), move up 4 units on the y-axis, and 2 units left on the x-axis to reach the end-point A(2,7).

Move (4 x 3 = 12 units) up on the y-axis, and (2 x 3 = 6 units) left on the x-axis tor reach the end-point of A'B'(-2, 15).

Similarly,

from C(4,3), move one unit up on the y-axis and one unit right on the x-axis, to reach point B(5,4).

From C(4,3), move (1 x 3 = 3 units) on the y-axis, (1 x 3 = 3 units) to the right on the x-axis, to reach the point B'(7,6).

New end-points: A'(-2, 15) and B'(7,6)

Find the length of bar (A'B'), using distance formula:

color(blue)(D = sqrt((x_2-x_1)^2+(y_2-y_1)^2)

rArr D=sqrt[(7-(-2)^2)+(6-15)^2)

rArr D=sqrt(9^2+(-9)^2)

rArr D=sqrt(162)

rArr D~~ 12.72792

bar (A'B')~~"12.73 units"

Hope this solution is helpful.