Question

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' ?

Answer 1

`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.