1. ## Conic Rotation

Hi

Please help me. I want to know how to find the coordinates of a circle after they have been rotated some degrees.

One problem I read wrote this:

"Shown in the standard (x,y) coordinate plane below, pint P (6,6) is rotated 90 degrees clockwise about the orgin (2,3). What are the coordinates of P after the rotation has been complete?"

I know the answer is (5,-1), but I don't know why that is the answer or how to solve the problem for other questions!

2. Hello klaus113

Welcome to Math Help Forum!
Originally Posted by klaus113
Hi

Please help me. I want to know how to find the coordinates of a circle after they have been rotated some degrees.

One problem I read wrote this:

"Shown in the standard (x,y) coordinate plane below, pint P (6,6) is rotated 90 degrees clockwise about the orgin (2,3). What are the coordinates of P after the rotation has been complete?"

I know the answer is (5,-1), but I don't know why that is the answer or how to solve the problem for other questions!
Think about any point whose coordinates are $\displaystyle (x_1,y_1)$ - see the attached diagram. Its horizontal and vertical distances from $\displaystyle (2, 3)$ are $\displaystyle x_1-2$ and $\displaystyle y_1-3$, respectively. After a rotation through $\displaystyle 90^o$ clockwise about this point, these will be the vertical distance below and the horizontal distance to the right of this point.

So the coordinates of the image of $\displaystyle (x_1,y_1)$ after this rotation are
$\displaystyle (2+y_1-3, 3 -[x_1-2])$; i.e. $\displaystyle (y_1-1, 5-x_1)$
This gives the general formula you're looking for, and confirms that $\displaystyle (6,6) \mapsto (5, -1)$.

3. Hello, klaus113!

Point $\displaystyle P(6,6)$ is rotated 90° clockwise about the point $\displaystyle Q(2,3)$.
What are the coordinates of $\displaystyle P$ after the rotation is completed?
A sketch is always a good idea . . .
Code:
        |         P
|           o (6,6)
|           :
|           :3
| Q o - - - +
|(2,3)  4
|
----+-----------------
|
Going from $\displaystyle Q$ to $\displaystyle P$, we move 4 right and 3 up.

Code:
        |
|(2,3) 3
| Q o - - +
|         :
|         :4
----+---------:----
|         o P'
|
Going from $\displaystyle Q$ to $\displaystyle P'$, we move 3 right and 4 down.
. .
Do you see why?

Now we know the location of $\displaystyle P'$.