8.12: Geometry Software and Graphing Rotations
- Page ID
- 6056
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Graph rotated images given preimage and number of degrees. Perform rotations using Geogebra.
Quadrilateral \(WXYZ\) has coordinates \(W(−5,−5)\), \(X(−2,0), \(Y(2,3)\) and \(Z(−1,3)\). Draw the quadrilateral on the Cartesian plane. Rotate the image \(110^{\circ}\) counterclockwise about the point \(X\). Show the resulting image.
Graphs of Rotations
In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A rotation is an example of a transformation where a figure is rotated about a specific point (called the center of rotation), a certain number of degrees.
For now, in order to graph a rotation in general you will use geometry software. This will allow you to rotate any figure any number of degrees about any point. There are a few common rotations that are good to know how to do without geometry software, shown in the table below.
Center of Rotation | Angle of Rotation | Preimage (Point \(P\)) | Rotated Image (Point \(P′\)) |
---|---|---|---|
\((0, 0)\) | \(90^{\circ}\) (or \(−270^{\circ} ) | \((x,y)\) | \((−y,x)\) |
\((0, 0)\) | \(180^{\circ}\) (or \(−180^{\circ}\) ) | \((x,y)\) | \((−x,−y)\) |
\((0, 0)\) | \(270^{\circ}\) (or \(−90^{\circ}\) ) | \((x,y)\) | \((y,−x)\) |
Let's draw the preimage and image and properly label each for the following transformation:
Line \(\overline{AB}\) drawn from \((-4, 2)\) to \((3, 2)\) has been rotated about the origin at an angle of \(90^{\circ}\) CW.
![f-d_4e8293b23f1108eb46b23b117bec739624ac7421dbdc195594564a7e+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3640/f-d_4e8293b23f1108eb46b23b117bec739624ac7421dbdc195594564a7e%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
Now, let's draw and label the rotated image for the following rotations:
- The diamond \(ABCD\) is rotated \(145^{\circ}\) CCW about the origin to form the image \(A′B′C′D′\).
![f-d_867d3e90e4b46c1dec081c2f1d13ec0ad1ebf473ee1eb5b37f3acc87+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3641/f-d_867d3e90e4b46c1dec081c2f1d13ec0ad1ebf473ee1eb5b37f3acc87%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
![f-d_15917e414351bed4facebc51b0fc65c528909378997202e4fdad295e+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3642/f-d_15917e414351bed4facebc51b0fc65c528909378997202e4fdad295e%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
Notice the direction is counter-clockwise.
- The following figure is rotated about the origin \(200^{\circ}\) CW to make a rotated image.
![f-d_6489c7a10df04071dccadd8d756f2c77bea669f2716b1ace4e0745a5+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3643/f-d_6489c7a10df04071dccadd8d756f2c77bea669f2716b1ace4e0745a5%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
![f-d_b1eb7fcd75f30bc2b8523d839684fe764a62ffebe2d35428a7cb3af8+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3644/f-d_b1eb7fcd75f30bc2b8523d839684fe764a62ffebe2d35428a7cb3af8%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
Notice the direction of the rotation is counter-clockwise, therefore the angle of rotation is \(160^{\circ}\).
Example \(\PageIndex{1}\)
Earlier, you were asked about the quadrilateral \(WXYZ\) has coordinates \(W(−5,−5)\), \(X(−2,0)\), Y(2,3)\) and \(Z(−1,3)\). Draw the quadrilateral on the Cartesian plane. Rotate the image \(110^{\circ}\) counterclockwise about the point X\). Show the resulting image.
Solution
![f-d_3ff27bec70435e89df832cb2242ab8fab65df5fa515f1b579891d4f9+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3645/f-d_3ff27bec70435e89df832cb2242ab8fab65df5fa515f1b579891d4f9%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
Example \(\PageIndex{2}\)
Line \(\overline{ST}\) drawn from \((-3, 4)\) to \((-3, 8)\) has been rotated \(60^{\circ}\) CW about the point \(S\). Draw the preimage and image and properly label each.
![f-d_9a247e66473af0d0e6b8cbca69dee2ac7587b495f2623199b2d49e8c+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3646/f-d_9a247e66473af0d0e6b8cbca69dee2ac7587b495f2623199b2d49e8c%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
Solution
Notice the direction of the angle is clockwise, therefore the angle measure is \(60^{\circ}\) CW or \(−60^{\circ}\).
Example \(\PageIndex{3}\)
The polygon below has been rotated \(155^{\circ}\) CCW about the origin. Draw the rotated image and properly label each.
![f-d_880a2f3c197e88037d4aff3df22000108b3429f09f91023a185cfbc2+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3647/f-d_880a2f3c197e88037d4aff3df22000108b3429f09f91023a185cfbc2%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
Solution
![f-d_63d6f804cf604a611bec938a0f61983a0efa7e00f1944ba3c6ecf663+IMAGE_THUMB_POSTCARD_TINY+IMAGE_THUMB_POSTCARD_TINY.png](https://k12.libretexts.org/@api/deki/files/3648/f-d_63d6f804cf604a611bec938a0f61983a0efa7e00f1944ba3c6ecf663%252BIMAGE_THUMB_POSTCARD_TINY%252BIMAGE_THUMB_POSTCARD_TINY.png?revision=1&size=bestfit&width=450)
Notice the direction of the angle is counter-clockwise, therefore the angle measure is \(155^{\circ}\) CCW or \(155^{\circ}\).
Example \(\PageIndex{4}\)
The purple pentagon is rotated about the point A \(225^{\circ}\). Find the coordinates of the purple pentagon. On the diagram, draw and label the rotated pentagon.
![f-d_d9f2c73ee75e6361c623ee4128c2fa68a44e0b7b743e5a2c6362a72e+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3658/f-d_d9f2c73ee75e6361c623ee4128c2fa68a44e0b7b743e5a2c6362a72e%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
Solution
![f-d_181564900f7f7aa9c2cf034dabec27754bc89bc96171f7435398dfbb+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3649/f-d_181564900f7f7aa9c2cf034dabec27754bc89bc96171f7435398dfbb%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
The measure of \(\angle BAB′=m\angle BAE′+m\angle E′AB′\). Therefore \(\angle BAB′=111.80^{\circ} +113.20^{\circ}\) or \(225^{\circ}\). Notice the direction of the angle is counter-clockwise, therefore the angle measure is \(225^{\circ}\) CCW or \(225^{\circ}\).
Review
![f-d_234afdf1317423c6f313c6e1127ac511da750b753685f77fa1caf1c1+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3650/f-d_234afdf1317423c6f313c6e1127ac511da750b753685f77fa1caf1c1%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
- Rotate the above figure \(90^{\circ}\) clockwise about the origin.
- Rotate the above figure \(270^{\circ}\) clockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.
![f-d_4f6233f7ca960b07c4aa34d7be4bc94232c3bb7afad60c012c860d61+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3651/f-d_4f6233f7ca960b07c4aa34d7be4bc94232c3bb7afad60c012c860d61%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
- Rotate the above figure \(90^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(270^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.
![f-d_00db51f117c77256de686a27748cdfa88c7f0ed0e633a59a3f89da32+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3652/f-d_00db51f117c77256de686a27748cdfa88c7f0ed0e633a59a3f89da32%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
- Rotate the above figure \(90^{\circ}\) clockwise about the origin.
- Rotate the above figure \(270^{\circ}\) clockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.
![f-d_7a0999108d4a8ddde2d12e9dbc80c751d5d6fdcec076d52a0c96c0dc+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3653/f-d_7a0999108d4a8ddde2d12e9dbc80c751d5d6fdcec076d52a0c96c0dc%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
- Rotate the above figure \(90^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(270^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.
![f-d_a0bd1ee4fe9000d0b093449e2ca6308318278563f9c95bda50d72a99+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3654/f-d_a0bd1ee4fe9000d0b093449e2ca6308318278563f9c95bda50d72a99%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
- Rotate the above figure \(90^{\circ}\) clockwise about the origin.
- Rotate the above figure \(270^{\circ}\) clockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.
![f-d_ca485b559bb107f944db21c19cb24bdd30b0eee9ee1f9fadc2416114+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3655/f-d_ca485b559bb107f944db21c19cb24bdd30b0eee9ee1f9fadc2416114%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
- Rotate the above figure \(90^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(270^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.
![f-d_ab7c6c52e913f82cdc35f16bafc69b8c2b49fbcbc9082c37fd4469b2+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3657/f-d_ab7c6c52e913f82cdc35f16bafc69b8c2b49fbcbc9082c37fd4469b2%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
- Rotate the above figure \(90^{\circ}\) clockwise about the origin.
- Rotate the above figure \(270^{\circ}\) clockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.
![f-d_b9dcf9e5f525629af5fa89480a949f75b148f600e1b7a4405d326c4e+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/3656/f-d_b9dcf9e5f525629af5fa89480a949f75b148f600e1b7a4405d326c4e%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
- Rotate the above figure \(90^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(270^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.
Review (Answers)
To see the Review answers, open this PDF file and look for section 10.8.
Vocabulary
Term | Definition |
---|---|
Rotation | A rotation is a transformation that turns a figure on the coordinate plane a certain number of degrees about a given point without changing the shape or size of the figure. |
Additional Resources
Practice: Geometry Software and Graphing Rotations