Expert: Sherry Wallin - 11/14/2009

Question
hi sherry

Please help me. I want to know how to find the coordinates of a point on the circumference of a circle after I have rotated it some degrees about the orgin.

One problem I faced was like this:

"as pictured below, point p (6,6) lies on the circumference of a circle with center (2,3). If the circle is rotated 90 degrees clockwise about its orgin, what are the new coordinates of point p?"

I found out that the answer is (5,-1) but I have no idea why that is the answer. Please help me deal with these types of problems!

Answer
Klaus~
    You are rotating clockwise 90 degrees, this means from the point (6,6) in Quadrant I you are moving down and over a certain number of units. Isn't this just the slope to the new point? But what do you know about the slope of a line? You know you can calculate it if you have two points and you also know that the slope of the second radius will be the negative reciprocal of the slope of the first radius. (perpendicular lines have negative reciprocal slopes). The first radius has slope m = (y2-y1)/(x2-x1) = (6-3)/(6-2) = 3/4. So the slope of the radius in the position 90 deg clockwise is -4/3. Your center will be the same, this means that the two radii intersect at the center (2,3). From there use your slope, i.e., go down 4 units and to the right 3 units, where does that put you? (2+3,3-4) = (5, -1).

Math Prof