Reflections

We all know what a reflection is. After all, we see our own reflections whenever we look in the mirror. But what does the term "reflection" mean in the world of math? While the general concept is the same, we need to cover some specific rules that apply only to geometrical reflection. Let's get started.

What is a reflection?

A reflection is a type of transformation representing a flip of a figure.

You might recall that when we transform a geometric shape, we simply change its shape and or position on a plane.

A reflection does not affect the size of the original shape, and it only affects its position.

When we reflect a figure, we flip it across some mirror line. This is the same concept as flipping a card upside down. But if the object is not symmetrical, it changes when we reflect it.

Because only the position changes, reflected images are "congruent" or equal to their original images.

How to create reflections

When we reflect figures, we must map every one of their points across a fixed line. We call this fixed line the "line of reflection."

If we have just one point to work with, reflections are simple:

A reflection of a point over the x -axis:

As you can see, this point has been reflected over the x-axis. You might have also spotted the fact that the reflective image now has a negative coordinate point. This is because when we reflect an image over the x-axis, we're always left with a negative y-value.

In other words, the rule for a reflection over the x -axis is:

$(x,y)becomes(x,-y)$ with a reflection over the x-axis.

Now let's see look at a reflection of a point over the y-axis:

We can see that a coordinate on the reflected image has become negative, but this time it's the x value instead of the y value. The rule in this case is $(x,y)becomes(-x,y)$.

But what about a reflection over a diagonal line? In other words, what if $y=x$? Take a look:

The rule is simple: We flip the two values. In other words, $(x,y)becomes(y,x)$

If $y=-x$, then the reflected values are both negative. In other words, $(x,y)becomes(-x,-y)$.

When are reflections useful?

We use reflections in many real-world applications:

• Manufacturing, especially in mirror-image objects like gloves, shoes, and spectacles
• Creating objects that need to be perfectly symmetrical, such as airplanes
• Chemists use reflection to create mirror images of sugar molecules, such as glucose and fructose

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