Line XY is dilated by a scale factor of 1.3 with the origin as the center of dilation to create the image line X'Y' . If the slope and length of XY are # m # and #l# respectively, what is the slope of X'Y' ?

1 Answer
Somebody N.
Jun 2, 2018

#color(blue)(m)#

Explanation:

If line xy has end points #(x_1,y_1),(x_2,y_2)#

Dilated line line has endpoints#((13x_1)/10,(13y_1)/10),((13x_2)/10,(13y_2)/10)#

#m=(y_2-y_1)/(x_2-x_1)#

After dilation:

#m=((13y_2)/10-(13y_1)/10)/((13x_2)/10-(13x_1)/10)=(13/10(y_2-y_1))/(13/10(x_2-x_1))=(y_2-y_1)/(x_2-x_1)#

If:

#l=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

After dilation:

#sqrt(((13x_2)/10-(13x_1)/10)^2+((13y_2)/10-(13y_1)/10)^2)#

#sqrt((13/10)^2(x_2-x_1)^2+(13/10)^2(y_2-y_1)^2)#

#sqrt((13/10)^2((x_2-x_1)^2+(y_2-y_1)^2))#

#13/10sqrt(((x_2-x_1)^2+(y_2-y_1)^2))#

#:.#

Dilated length:

#13/10l#

In the above, the work carried out to find the slope of the dilated line was unnecessary. Dilations do not change the orientation, so the gradient would remain the same.

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