Question

point f has coordinated (8,-1), point g is symmetric to point f with respect to the line y=x, what are the coordinates of point g

Answer 1

If points f and g are symmetric with respect to the line y=x, then the line connecting f and g is perpendicular to y=x, and f and g are equidistant from y=x.

This problem could be solved graphically by graphing y=x and (8,-1).  With a ruler, measure the perpendicular distance from y=x of (8,-1), and then plot point g that distance from y=x in the opposite direction.  Read the coordinates of point g from the graph.

Alternatively, calculate the distance from y=x of (8,-1).  As before, this distance is perpendicular to y=x and is measured along the line y= -x + b, where b is the vertical intercept of this line.  What is b?  y = -x + b must be satisfied by (8,-1):  -1 = -8 + b, or b = 7.  Then the line thru (8,-1) perpendicular to y=x is     y = -x + 7.  Where does this line intersect y = x?

y = x = y = -x + 7, or 2x = 7, or x = 3.5.  Since y=x, the point of intersection of y=x and y= -x + 7 is (3.5, 3.5).

Use the distance formula to determine the distance between (3.5, 3.5) and (8, -1).  This produces the answer to this question.