Specify the coordinates of the point symmetric to the given point K (19.3) (9.75).

Obviously, a point symmetric to a given point relative to point K will lie on the same straight line with these points.

Let us compose the equation of the straight line passing through the points (19; 3) and
(9; 75):

3 = 19k + b; (one)

75 = 9k + b;

Subtract the second from the first equation:

3 – 75 = 19k + b – 9k – b;

10k = – 72;

k = – 7.2;

Find b by substituting k in (1):

3 = 19 * (- 7.2) + b;

b = 3 + 136.8;

b = 139.8.

So, the equation of the straight line passing through the points (19; 3) and
(9; 75): y = – 7.2x + 139.8.

The point (9; 75) is located to the left of the point (19; 3), which means that the point symmetric to it is located to the right of the point (19; 3) at the same distance along the x axis, that is, its abscissa is 19 + (19 – 9) = 29.

Find the ordinate by substituting the abscissa into the equation of the straight line:

y = – 7.2 * 29 + 139.8 = – 69.

Answer: (29; – 69).