Geometry- What is center-point?

Question

This is the question: Given PQ with P(–2, 2) and Q(4, 4) is rotated 90 degrees counterclockwise with a center of rotation at A(2, –1), graph and list the coordinates of P'Q' .

What are the new coordinates, and how can I find them?

Answer 1

Let's look at what happens to a point that is rotated about the origin cc 90 degrees:

(x,y) --> (-y,x) If you draw this with any point (x,y) in the 1st quadrant, you see that the angle between a line from the origin to the point and with the x- axis is same as the vertical angle after the rotation made with the y axis. This means that x and y coordinates have switched and that the new "x" is negative (2nd Quadrant)

Now rotating around another point than (0,0) means adjusting your coordinate system to a new origin (h,k) = (2,-1):

(x,y) --> (-(y-k), x-h) or (-2,2) --> (-(2+1), -2-2) = (-3,-4)

The other point (4,4) transformation, I leave to you. All points of the rotated line segment will be between these two transformed points.

Good luck.

Answer 2

To me, the easiest way to do this problem is to create a new coordinate plane located with the origin at the rotation point (2, -1). Then reassign the points (-2, 2) and (4, 4) new coordinates based on the new origin like this:

Then to rotate 90°, just switch the x and y values (but you might need to adjust the positive/negative signs accordingly).

Rotating the new point (-4, 3) (was (-2, 2)) 90° CW would put it in quadrant 1 so both x and y would be positive, so the new coordinate point would be (3, 4). Now, adjust that back to the original coordinate system by adding 2 to the x-value and subtracting 1 from the y value to get (5, 3), So P' is (5, 3).

Rotating the new point (2, 5) (was (4, 4)) 90° CW would put it in quadrant 4 so the x would be positive and the y would be negative, so the new coordinate point would be (5, -2). Now, adjust that back to the original coordinate system by adding 2 to the x-value and subtracting 1 from the y value to get (7, -3), So Q' is (7, -3).

Answer 3

Hello Shelly,

I don't know if you use formulas in your class. You can do it without formulas if you draw a graph.

Plot the 3 points.

I will tell you how to rotate P, and then see if you can rotate Q.

First I see how I would travel to P from A. I would go left 4 and up 3. Where would point P be after I rotate 90^o counterclockwise? In the 3rd quadrant somewhere. In a 90^o rotation, the left/right number will become the up/down number and the up/down number will become the left/right number.

To go from the center to the 3rd quadrant, I will need to go left and down. I must switch the two numbers so it is left 3 and down 4 from the center (2, -1). Where would that be? (-1, -5) so that is P'.

See if you can find Q'.