New York Geometry - 2020 Edition

4.02 Reflections

Lesson

A reflection occurs when we flip an object or shape across a line, like in a mirror. The image of the reflection is congruent to the preimage, just flipped in position. Every point on the object or shape has a corresponding point on the image, and they will both have the same distance from the reflection line. We can reflect points, lines, polygons on the $xy$`x``y`-plane by flipping them across an axis line or another line in the plane.

If we reflect horizontally across the $y$`y` axis, then the $y$`y`-values of the coordinates remain the same and the $x$`x` values change sign.

Note how the point $\left(-2,1\right)$(−2,1) becomes $\left(2,1\right)$(2,1). The $y$`y`-values have not changed and the $x$`x`-values have changed signs.In this diagram, the image is reflected across $y$`y` axis.

Similarly the point $\left(-6,3\right)$(−6,3) becomes $\left(6,3\right)$(6,3). The $y$`y`-values have not changed and the $x$`x`-values have changed signs.

If we reflect vertically across the $x$`x` axis, then the $x$`x` values of the coordinates remain the same and the $y$`y` values change sign.

Note how the point $\left(4,3\right)$(4,3) becomes $\left(4,-3\right)$(4,−3). The $x$`x` values have not changed and the $y$`y` values have changed signs.In this diagram, the image is reflected across $x$`x` axis.

Similarly the point $\left(0,5\right)$(0,5) becomes $\left(0,-5\right)$(0,−5). The $x$`x` values have not changed and the $y$`y` values have changed signs.

Use this interactive below to further consolidate the ideas behind translations on the $xy$`x``y`-plane.

Slide the right-most point to change the $y$`y`-intercept and/or the slope of the line of reflection. Slide points on the Object to change the shape of the preimage triangle.

What do you notice about the distance to the reflection line for corresponding points?

What happens when you reflect in the line $y=x$`y`=`x`? $y=-x$`y`=−`x`?

Consider the point $A\left(7,3\right)$`A`(7,3).

Plot point $A$

`A`on the number plane.Loading Graph...Now plot point $A'$

`A`′, a reflection of point $A$`A`across the $x$`x`-axis.Loading Graph...

Consider the point $A\left(-7,-3\right)$`A`(−7,−3).

Plot point $A$

`A`on the number plane.Loading Graph...Now plot point $A'$

`A`′, a reflection of point $A$`A`across the $y$`y`-axis.Loading Graph...

Consider the line segment $AB$`A``B`, where the endpoints are $A$`A`$\left(-4,-2\right)$(−4,−2) and $B$`B`$\left(6,7\right)$(6,7).

Plot the line segment $AB$

`A``B`on the number plane.Loading Graph...Now plot the reflection of the line segment $AB$

`A``B`across the $x$`x`-axis.Loading Graph...

Plot the new triangle formed by reflecting the given triangle about the line $x=-3$`x`=−3.

- Loading Graph...

Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not.

Develop definitions of rotations, reflections, and translations in terms of points, angles, circles, perpendicular lines, parallel lines, and line segments.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto another.