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12.7: Chapter 12 Review

Difficulty Level: At Grade Created by: CK-12

Keywords & Theorems

Line of Symmetry
A line that passes through a figure such that it splits the figure into two congruent halves.
Line Symmetry
When a figure has one or more lines of symmetry.
Rotational Symmetry
When a figure can be rotated (less that \begin{align*}360^\circ\end{align*}) and it looks the same way it did before the rotation.
Center of Rotation
The point at which the figure is rotated around such that the rotational symmetry holds. Typically, the center of rotation is the center of the figure.
angle of rotation
The angle of rotation, tells us how many degrees we can rotate a figure so that it still looks the same.
Transformation
An operation that moves, flips, or changes a figure to create a new figure.
Rigid Transformation
A transformation that preserves size and shape.
Translation
A transformation that moves every point in a figure the same distance in the same direction.
Vector
A quantity that has direction and size.
Reflection
A transformation that turns a figure into its mirror image by flipping it over a line.
Line of Reflection
The line that a figure is reflected over.
Reflection over the \begin{align*}y-\end{align*}axis
If \begin{align*}(x,y)\end{align*} is reflected over the \begin{align*}y-\end{align*}axis, then the image is \begin{align*}(-x,y)\end{align*}.
Reflection over the \begin{align*}x-\end{align*}axis
If \begin{align*}(x,y)\end{align*} is reflected over the \begin{align*}x-\end{align*}axis, then the image is \begin{align*}(x,-y)\end{align*}.
Reflection over \begin{align*}x = a\end{align*}
If \begin{align*}(x, y)\end{align*} is reflected over the vertical line \begin{align*}x = a\end{align*}, then the image is \begin{align*}(2a - x, y)\end{align*}.
Reflection over \begin{align*}y = b\end{align*}
If \begin{align*}(x, y)\end{align*} is reflected over the horizontal line \begin{align*}y = b\end{align*}, then the image is \begin{align*}(x, 2b - y)\end{align*}.
Reflection over \begin{align*}y = x\end{align*}
If \begin{align*}(x, y)\end{align*} is reflected over the line \begin{align*}y = x\end{align*}, then the image is \begin{align*}(y, x)\end{align*}.
Reflection over \begin{align*}y = -x\end{align*}
If \begin{align*}(x, y)\end{align*} is reflected over the line \begin{align*}y = -x\end{align*}, then the image is \begin{align*}(-y, -x)\end{align*}.
Rotation
A transformation by which a figure is turned around a fixed point to create an image.
Center of Rotation
The fixed point that a figure is rotated around.
Rotation of \begin{align*}180^\circ\end{align*}
If \begin{align*}(x, y)\end{align*} is rotated \begin{align*}180^\circ\end{align*} around the origin, then the image will be \begin{align*}(-x, -y)\end{align*}.
Rotation of \begin{align*}90^\circ\end{align*}
If \begin{align*}(x, y)\end{align*} is rotated \begin{align*}90^\circ\end{align*} around the origin, then the image will be \begin{align*}(-y, x)\end{align*}.
Rotation of \begin{align*}270^\circ\end{align*}
If \begin{align*}(x, y)\end{align*} is rotated \begin{align*}270^\circ\end{align*} around the origin, then the image will be \begin{align*}(y, -x)\end{align*}.
Composition (of transformations)
To perform more than one rigid transformation on a figure.
Glide Reflection
A composition of a reflection and a translation. The translation is in a direction parallel to the line of reflection.
Reflections over Parallel Lines Theorem
If you compose two reflections over parallel lines that are \begin{align*}h\end{align*} units apart, it is the same as a single translation of \begin{align*}2h\end{align*} units.
Reflection over the Axes Theorem
If you compose two reflections over each axis, then the final image is a rotation of \begin{align*}180^\circ\end{align*} of the original.
Reflection over Intersecting Lines Theorem
If you compose two reflections over lines that intersect at \begin{align*}x^\circ\end{align*}, then the resulting image is a rotation of \begin{align*}2x^\circ\end{align*}, where the center of rotation is the point of intersection.
Tessellation
A tiling over a plane with one or more figures such that the figures fill the plane with no overlaps and no gaps.

Review Questions

Match the description with its rule.

  1. Reflection over the \begin{align*}y-\end{align*}axis - A. \begin{align*}(2a - x, y)\end{align*}
  2. Reflection over the \begin{align*}x-\end{align*}axis - B. \begin{align*}(-y, -x)\end{align*}
  3. Reflection over \begin{align*}x = a\end{align*} - C. \begin{align*}(-x, y)\end{align*}
  4. Reflection over \begin{align*}y = b\end{align*} - D. \begin{align*}(-y, x)\end{align*}
  5. Reflection over \begin{align*}y = x\end{align*} - E. \begin{align*}(x, -y)\end{align*}
  6. Reflection over \begin{align*}y = -x\end{align*} - F. \begin{align*}(x, 2b - y)\end{align*}
  7. Rotation of \begin{align*}180^\circ\end{align*} - G. \begin{align*}(x, y)\end{align*}
  8. Rotation of \begin{align*}90^\circ\end{align*} - H. \begin{align*}(-x, -y)\end{align*}
  9. Rotation of \begin{align*}270^\circ\end{align*} - I. \begin{align*}(y, -x)\end{align*}
  10. Rotation of \begin{align*}360^\circ\end{align*} - J. \begin{align*}(y, x)\end{align*}

Texas Instruments Resources

In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9697.

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CK.MAT.ENG.SE.1.Geometry-Basic.12.7