1. ## Transformation geometry

Hi everyone. I'm confused with determining the general rule when a figure has been rotated NOT at 90, 180, 270, 360 degrees.

For example: In triangle ABC: A(4;4), B(4;1) C(2;0) has changed, C'(-1.5;1.5)... A'(-5,6;0) and B'(-3,6;3) NB - (not sure about A' and B' they approximations)

So how would i determine the general rule. How do i go about this? I know you use the formula:

(x;y) → (xCosθ - ySinθ; yCosθ + xSinθ)
Can i get some step by step help please... thnx

2. Hello,
Originally Posted by pozmans
Hi everyone. I'm confused with determining the general rule when a figure has been rotated NOT at 90, 180, 270, 360 degrees.

For example: In triangle ABC: A(4;4), B(4;1) C(2;0) has changed, C'(-1.5;1.5)... A'(-5,6;0) and B'(-3,6;3) NB - (not sure about A' and B' they approximations)

So how would i determine the general rule. How do i go about this? I know you use the formula:

(x;y) → (xCosθ - ySinθ; yCosθ + xSinθ)
Can i get some step by step help please... thnx
But the transformation from ABC to A'B'C' is not a rotation

3. What kind of transformation has occured then?

4. Well, I drew the points. And I drew the triangles.
And I don't see any simple transformation from that...

5. Originally Posted by pozmans
What kind of transformation has occured then?
To me, it looks approximately like a rotation around (0,0), with angle 135°(= $\frac{3\pi}{4}$ rad).

Your formula (x;y) → (xCosθ - ySinθ; yCosθ + xSinθ) is correct.

You can prove it in various ways. For instance, you may know that you can write $x=r\cos\alpha$ and $y=r\sin\alpha$ for some $r>0$ and $\alpha\in\mathbb{R}$ (this is polar coordinates). Then $(x,y)$ is rotated into $(r\cos(\alpha+\theta),r\sin(\alpha+\theta))$. If you expand the sin and cos using the usual formulas, you'll get the expected formula.