10.6: Rotating
Learning Objectives
 Find the image of a point in a rotation in a coordinate plane.
 Recognize that a rotation is an isometry.
Sample Rotations
In this lesson we will study rotations centered at the origin of a coordinate plane. We begin with some specific examples of rotations. Later we will see how these rotations fit into a general formula.
We define a rotation as follows: In a rotation centered at the origin with an angle of rotation of
The original preimage point is one endpoint of the arc, and the image of the original point is the other endpoint of the arc:
 Rotations centered at the origin move points ______________________ along an arc of a circle.
 For a rotation of
n∘ , the central angle of the circle measures ____________.  The preimage point is one endpoint of the _________________ and the image is the other endpoint.
180∘ Rotation
Our first example is rotation through an angle of
In a
Notice:

P andP′ are the endpoints of a diameter of a circle.
The distance from
 The rotation is the same as a “reflection in the origin.”
A
In a rotation of
A
Use the Distance Formula:
So
 A
180∘ rotation is an ___________________________, so distance is preserved.  When a segment is rotated
180∘ (or reflected in the origin), its image is a _________________________________ segment.
90∘ Rotation
The next example is a rotation through an angle of
In a
Notice:

PO¯¯¯¯¯ andP′O¯¯¯¯¯¯ are both radii of the same circle, soPO=P′O .
If

∠POP′ is a right angle.  The acute angle formed by
PO¯¯¯¯¯ and thex− axis and the acute angle formed byP′O¯¯¯¯¯¯ and thex− axis are complementary angles.
Remember, complementary angles add up to ________________
You can see by the coordinates of the preimage and image points, in a
 the
x− andy− coordinates are switched AND  the
x− coordinate is negative.
In a
A
Use the Distance Formula:
So \begin{align*}PQ = P^\prime Q^\prime\end{align*}
Reading Check:
Which of the following are isometries? Circle all that apply:
\begin{align*}& 30^\circ rotation && 45^\circ rotation && 60^\circ rotation\\ & 90^\circ rotation && 150^\circ rotation && 180^\circ rotation\\ & Reflection && Translation && Bisection\end{align*}
Example 1
What are the coordinates of the vertices of \begin{align*}\Delta ABC\end{align*} in a rotation of \begin{align*}90^\circ\end{align*}?
Point \begin{align*}A\end{align*} is (4, 6), \begin{align*}B\end{align*} is (–4, 2), and \begin{align*}C\end{align*} is (6, –2).
In a \begin{align*}90^\circ\end{align*} rotation, the \begin{align*}x\end{align*}coordinate and the \begin{align*}y\end{align*}coordinate are switched AND the new \begin{align*}x\end{align*}coordinate is made negative:
 \begin{align*}A\end{align*} becomes \begin{align*}A^\prime\end{align*} : switch \begin{align*}x\end{align*} and \begin{align*}y\end{align*} to (6, 4) and make \begin{align*}x\end{align*} negative (–6, 4)
 \begin{align*}B\end{align*} becomes \begin{align*}B^\prime\end{align*} : switch \begin{align*}x\end{align*} and \begin{align*}y\end{align*} to (2, –4) and make \begin{align*}x\end{align*} negative (–2, –4)
 \begin{align*}C\end{align*} becomes \begin{align*}C^\prime\end{align*} : switch \begin{align*}x\end{align*} and \begin{align*}y\end{align*} to (–2, 6) and make \begin{align*}x\end{align*} negative \begin{align*}( (2), 6) = (2, 6)\end{align*}
So the vertices of \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*} are (–6, 4), (–2, –4), and (2, 6).
Plot each of these points on the coordinate plane above and draw in each side of the new rotated triangle. Can you see how \begin{align*}\Delta ABC\end{align*} is rotated \begin{align*}90^\circ\end{align*} to \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*}?
Reading Check:
1. True or false: A rotation is always in the counterclockwise direction.
2. On the coordinate plane below, create a point anywhere you like, and label it \begin{align*}P\end{align*}.
Then draw a second point \begin{align*}W\end{align*} that is the image of point \begin{align*}P\end{align*} rotated \begin{align*}180^\circ\end{align*}.
3. On the coordinate plane above, draw a third point \begin{align*}R\end{align*} that is the image of your original point \begin{align*}P\end{align*} rotated \begin{align*}90^\circ\end{align*}.
4. Is a \begin{align*}90^\circ\end{align*} rotation an isometry? Explain.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
5. Is a \begin{align*}180^\circ\end{align*} rotation an isometry? Explain.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
6. What type of rotation is the same as a reflection in the origin?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
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Date Created:
Feb 23, 2012Last Modified:
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