Rotations/Reflections of a point

Question

It's been $10$ years since I see some kind of geometry and I'm preparing for a test of these sort of questions. I need help figuring out the following problems:

1) A point $P(x,y)$ is rotated $180$ degrees about the origin, then reflected over the y-axis. What is the resulting image of $P$?

My conjecture: $(-x,y)$ but I don't quite agree with it graphically. I'm having trouble with what exactly a 180 degree rotation about the origin simply is.

2) A point $P(x,y)$ is reflected over the y-axis and then rotated 180 degrees about the origin. What is the resulting image?

Is this the same thing as 1?

3) A point $A(x,y)$ is reflected over the line $y=x$ and then reflected over the y-axis. What is the resulting image of A?

My conjecture: $(-y,x)$

4) A point $A(x,y)$ is reflected over the lines $y=-x$ and then reflected over the y-axis. What is the resulting image of A?

My conjecture: $(y,-x)$

Answer 1

If a point $P(a,b)$ is rotated $180$ degree about the origin, then the resulting image of $P$ is $(−a,−b)$.

1) $(x,y)\rightarrow (-x,-y)\rightarrow (x,-y).$

2) $(x,y)\rightarrow (-x,y)\rightarrow (x,-y).$

Hence, this is the same as 1.

3) $(x,y)\rightarrow (y,x)\rightarrow (-y,x).$

4) $(x,y)\rightarrow (-y,-x)\rightarrow (y,-x).$

Hence, your conjectures are correct.

Answer 2

(1)If $P(x_1,y_1)$ is rotated angle $\phi$ about $O(a,b)$ to $P'(x',y')$

we have $|OP|=|OP'|$ and $|\arctan m_\text{OP}-\arctan m_\text{OP'}|=\phi$

By atan$(z)$ I imply atan2

(2)If $P(x_1,y_1)$ is reflected over $L:ax+by+c=0$ and the resulting image of $P$ is $Q(x_2,y_2)\ \ \ \ (1)$

We have $\displaystyle R\left(\frac{x_1+x_2}2,\frac{y_1+y_2}2\right) $ will lies on $(1)$

and $\displaystyle PQ\perp L\implies \frac{y_2-y_1}{x_2-x_1}=m_{\text{L}}=-\frac ab$