How Do Reflections Along Different Lines Affect Point Coordinates?

[blafu]

We were asked to make generalisations for the following:

a.) if a point is reflected along y = x+c
b.) if a point is reflected along y = -x+c
i came up w/ the ff.:
a.) if a point is reflected along y = x+c, then the general point (a, b) will become (b-c, a+c)

b.) if a point is reflected along y = -x+c, then the general point (a, b) will become (-b+c, -a+c)

let me knw if these are correct... if not then pls guide me and tell me the correct generalisations thank u!

 

[HallsofIvy]

One easy way to check: if points are reflected around a line, then points on the line are not changed. (0, c) is on the line. If "(a, b) will become (b- c, a+ c)" then (0, c) will become (0, c). That looks good. More generally, (x, x+ c) will become ((x+c)-c, x+ c)= (x, c).

If y= -x+ c then (x, -x+ c), using (-b+c, -a+c) becomes (-(-x+c)+ c, -x+ c)= (x, -x+c) again.

To complete the check, to see if the line from (a, b) to (b-c, a+ c) is always perpendicular to y= x+ c. The slope of the line from (a, b) to (b-c, a+ c) is (a+c- b)/(b-c-a)= -1. The slope of y= x+ c is 1. Yes they are perpendicular.

 

[blafu]

Thank you very much...just a few followup questions (I hope it is ok to ask in this thread as well):

1) what would happen to point (a, b) if it is reflected along y = mx+c, where m is not 1 or -1 (that is, a line which slope is not -1 or 1)?
2) can "when a point A is reflected along line y = -mx+c, the line made by points A and A' have slope of 1" be made as a generalisation too?
3) can "when a point B is reflected along line y = mx+c, the line made by points B and B' have slope of -1" be made as a generalisation too?

for 2 and 3, provided that m is not 0 or undefined..., and that A and A', and B and B' are not invariant points.