什么是对称变换（相对于一条直线的）

A reflection is colloquially known as a flip because it does the same thing a mirror does $-$ flips an object over a line or point or plane into an image.

For example, if a point $A$ is reflected over a line $\ell$ to a point $A'$, we have that the distance between $A$ and $\ell$ is the same as the distance between $A'$ and $\ell$. The same holds for sets of points such as lines and planes.

Intuitively, we think that reflections are isometries. After all, when we look in the mirror, it seems as though the reflection is our size. So how do we prove this?

Suppose that the segment $AB$ is reflected about a flat $n$-dimensional (don't worry, the value of $n$ doesn't really matter) surface $S$ to the segment $A'B'$. We would like to prove that $AB=A'B'$, regardless of the surface.

Suppose that $AA'$ meets $S$ at $X_A$ and that $BB'$ meets $S$ at $X_B$.

From the definition of reflection, we have the following statements: $$AX_A=A'X_A$$ $$BX_B=B'X_B$$ $$AA'\perp S$$ $$BB'\perp S$$

Of course, $X_AX_B$ is congruent to itself.

Now look at the quadrilaterals $AX_AX_BB$ and $A'X_AX_BB'$.

Clearly, these two quadrilaterals are congruent, since the remaining parts of the quadrilateral may be completely determined from the existing information.

Thus, since $AB$ and $A'B'$ are corresponding segments in the two quadrilaterals, we have $AB=A'B'$.

Thus, a reflection preserves distances, so it is an isometry.

(Video) Reflection

by Planet Nutshell

Remember this?

Who is that girl I see $\\$ Staring straight back at me? $\\$ When will my reflection show $\\$ Who I am inside? $\\$

Yep, it's from the song Reflection, featured in the movie Mulan. The use of reflection here is much more poetic--Mulan sees herself in the reflection, yet she is not seeing herself.

Reflection works slightly differently in geometry. It can be thought of as "folding or "flipping" an object over the line of reflection." (source) $\\$

Here is a video that goes deeper into reflection.

The video explains reflection and some properties of reflection. Try this problem to test your understanding of reflection.

(Video) How Do I Reflect a Figure

by MashUp Math

Here is a video talking about the properties of reflections.

Here is a brief summary of the video:

A reflection is a mirror image. It is NOT a change in size or shape.$\\$ The line of reflection, or line of symmetry, can be any line you choose, but the x and y axis are some common lines of reflection.

Notation$\\$ $r_{equation}$ is used to denote the line of symmetry. Replace "equation" with any line--y=x, y=0, x=1, etc.

The distance of a point to the line of symmetry is the same as the distance of its reflection to the line.

Watch the following clip to learn about reflection of a line segment and a figure.

The above clip shows some examples of reflecting line segments and geometric figures. Try doing them yourself to get better at it!

Try the problem below to test your understanding.