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Here are a few recommended readings before getting started with this lesson.
$PA$ | $PA_{′}$ | ||
---|---|---|---|
$PB$ | $PB_{′}$ | ||
$PC$ | $PC_{′}$ | ||
$PD$ | $PD_{′}$ |
$m∠APA_{′}$ | |
---|---|
$m∠BPB_{′}$ | |
$m∠CPC_{′}$ |
A rotation is a transformation in which a figure is turned about a fixed point $P.$ The number of degrees the figure rotates $α_{∘}$ is the angle of rotation. The fixed point $P$ is called the center of rotation. Rotations map every point $A$ in the plane to its image $A_{′}$ such that one of the following statements is satisfied.
Rotations can be performed by hand with the help of a straightedge, a compass, and a protractor.
To rotate point $A$ about point $P$ by an angle of $130_{∘}$ measured counterclockwise, follow these five steps.
Place the center of the protractor on $P$ and align it with $PA.$
The protractor is placed as illustrated above when the rotation is counterclockwise. If the rotation has to be done clockwise, the protractor needs to be placed as follows.
Locate the corresponding measure on the protractor and make a small mark. In this case, the mark will be made at $130_{∘}.$
Notice that this method of construction has also confirmed that $PA$ is congruent to $PA_{′}.$
Start by rotating the vertices of the triangle. To do so, use a protractor to draw a $180_{∘}$ angle.
Zosia's goal is to find the image of the triangle after a $180_{∘}$ clockwise rotation about its fixed vertex. Labeling the vertices of the triangle will make this rotation process simpler.
The center of rotation is $O$ and the angle of rotation is $180_{∘}.$ Perform the rotation by rotating one point at a time. To rotate $N,$ place the center of the protractor on $O$ and align it with $NO.$ Use the protractor to draw a ray that starts from $O$ and makes a $180_{∘}$ angle with $NO.$
Then, mark a point $N_{′}$ on this ray so that $ON_{′}$ is the same length as $ON.$ This is the image of $N$ after the rotation. Since $ON$ is the radius of the inner circle, $N_{′}$ should also be on that circle.
Repeat the same process for the vertex $E$ to find $E_{′}.$ Since $O$ is the center of rotation, $O_{′}$ will be in the same position as $O.$
Finally connect $N_{′},$ $E_{′},$ and $O_{′}$ to draw the image of $NEO$ after the $180_{∘}$ rotation.
As shown, the image $N_{′}$ corresponds to P and $E_{′}$ corresponds to T. Therefore, the clue Vincenzo and Zosia is looking for is NEPT.
Counterclockwise Rotations Around the Origin | |
---|---|
Angle of Rotation | Rule |
$90_{∘}$ | $(x,y)→(-y,x)$ |
$180_{∘}$ | $(x,y)→(-x,-y)$ |
$270_{∘}$ | $(x,y)→(y,-x)$ |
The clue Vincenzo and Zosia found reminded them of the word Neptune.
Vincenzo had previously heard stories about this magical place. He had even heard about three ordinary students who had found a door into a mystical library. Could this be another doorway to that place? As they got closer, a puzzle appeared between the columns of the gate.
When a point with coordinates $(x,y)$ is rotated $90_{∘}$ counterclockwise about the origin, its image becomes $(-y,x).$
Zosia and Vincenzo want to rotate the figure $90_{∘}$ counterclockwise around the origin. This can be achieved by rotating each point of the original figure. In this case, four points will be sufficient to determine the position of the image.
When a counterclockwise rotation is performed about the origin, the coordinates of the image can be written in relation to the coordinates of the preimage.
Counterclockwise Rotations About the Origin | ||
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$90_{∘}$ Rotation | $180_{∘}$ Rotation | $270_{∘}$ Rotation |
$Preimage(x,y) → Image(-y,x) $ |
$Preimage(x,y) → Image(-x,-y) $ |
$Preimage(x,y) → Image(y,-x) $ |
As they explore the magical realm, Vincenzo and Zosia come across Dilatius the Dimension Shifter, a wizard who can change the size of objects using dilations. They realize that the notebook they found belongs to the wizard and excitedly ask him to teach them about dilation. Dilatius thrilled to share his knowledge with them.
$OA_{′}=k⋅OA⇔k=OAOA_{′} $
When a point is dilated using a scale factor of $k$ and a center of dilation at the origin, the coordinates of its image are found by multiplying the coordinates of the preimage by $k.$
$(x,y)→(kx,ky)$
The diagram shows how the image changes as the preimage and the scale factor change.
Dilatius is impressed by Vincenzo and Zosia's eagerness to learn. They seem to have picked up the dilation spell using the coordinate rule quickly, so he challenges them to dilate the following triangle.
Draw the image of the triangle after a dilation with center $(0,0)$ and a scale factor of $3.$To find the image of a vertex after a dilation with scale factor $k,$ multiply its coordinates by $k.$
Zosia and Vincenzo need to dilate the triangle using a scale factor of $3$ with respect to the origin. Start by identifying the vertices of the triangle.
When the center of dilation is the origin, each coordinate of the preimage is multiplied by the scale factor $k$ to find the coordinates of the image.Dilation With Scale Factor $k=3$ | ||
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Preimage | Multiply by $k$ | Image |
$A(0,2)$ | $(3⋅0,3⋅2)$ | $A_{′}(0,6)$ |
$B(3,1)$ | $(3⋅3,3⋅1)$ | $B_{′}(9,3)$ |
$C(2,-1)$ | $(3⋅2,3⋅(-1))$ | $C_{′}(6,-3)$ |
Dilatius teaches Vincenzo and Zosia to use reducio
to make things smaller and enlargio
to make them bigger. These phrases produce a reduction and an enlargement, respectively. He then quizzes them about the magic behind the drawings in his notebook.
The scale factor is the ratio of the sides lengths of the image to the corresponding side lengths of the original figure.
There are two types of dilations.
In the given coordinate plane, it can be seen that the green square is smaller than the blue square. This means that the dilation is a reduction.
Next, remember that the scale factor is the ratio of the sides lengths of the image to the corresponding side lengths of the preimage.