The point A(-6,6) is reflected over the point (0,7) and its image is point B. What are the coordinates of point B?

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Explanation:

Let's label (0,7) as point C.

Plot points A and C on the same coordinate system.

Note that to go from A to C, we must do two things (in either order)

These two items will be replicated in either order to go from point C to point B.

The translation rule we follow is which is a mathematical way of saying "go right 6 and up 1". If we plug in x = -6 from point A we get to x = 0 which is the x coordinate of point C. Repeating this leads to x = 6, which is the x coordinate of point B.

Through similar arguments, we go from y = 6, to y = 7, to y = 8, when we hop from points A, to C, to B in that exact order.

Put everything together and we see that B is located at (6,8)

The diagram below provides a visual summary.

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Extra info:

If we draw a line from point A to B, then point C will be on this line. Furthermore, AC = CB which can be proven using congruent triangles. Reflections preserve distance, meaning the distance from A to the mirror point C, is the same as from B to C as well. In short, we can say C is the midpoint of segment AB.