Final answer:
To find the image of a point (h,k) after a half turn around point A(2,0), we translate the point, perform the rotation, and translate back, resulting in the final coordinates of (2-h, -k).
Step-by-step explanation:
The question asks for the coordinates of the image of a point with coordinates (h,k) after performing a half turn or rotation of 180 degrees about another point A with coordinates (2,0). A half turn about a point means that the image of any point after this transformation will be located directly opposite it on a circle centered at the point of rotation, here being point A. The distance from the point to the center of rotation remains the same before and after the rotation.
Let's say we have a point P with coordinates (h,k). The first step is to translate point P by subtracting the coordinates of point A, so we get a temporary point P' with coordinates (h-2, k-0) or (h-2, k). Next, we perform the half turn about the origin (after translation) which results in the point (-(h-2), -k). Finally, we translate this point back to the original position by adding the coordinates of point A, so we get the image point P'' with coordinates (-(h-2)+2, -k+0) or (2-h, -k). This is the image of the original point P after a half turn around point A.