Rotations
A rotation is a transformation in which a figure is turned about a fixed point. The fixed point is the center of rotation. Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation.
Rotations can be clockwise or counterclockwise as shown below. Counterclockwise rotations are positive angle measures and clockwise rotations are negative angle measures. In the Digi, all rotations, unless noted, will be counterclockwise.
A rotation is a transformation in which a figure is turned about a fixed point. The fixed point is the center of rotation. Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation.
Rotations can be clockwise or counterclockwise as shown below. Counterclockwise rotations are positive angle measures and clockwise rotations are negative angle measures. In the Digi, all rotations, unless noted, will be counterclockwise.
A rotation about point \(K\) through \(x\) degrees is a transformation that maps every point \(A\) in the plane to to a point \(A’\). so that the following properties are true:
A rotation is an isometry. This means that the size and shape stay the same after the rotation is performed.
- If \(A\) is not at the center of rotation, \(m\angle AKA' = x^{\circ}\).
- If \(A\) is at the center of rotation, then \(A = A’\).
- \(AB = A'B'\)
A rotation is an isometry. This means that the size and shape stay the same after the rotation is performed.
Rotate a figure in a Coordinate Plane
When we rotate a figure around the origin, we can simply take our paper and rotate it in the direction that we need to. Let’s look at the example below.
Example 1:
Given \(\triangle ROT\) has vertices \(R(2, 3), O(6, 1)\) and \(T(5, 4)\), rotate it \(90^{\circ}\) counterclockwise around the origin.
When we rotate a figure around the origin, we can simply take our paper and rotate it in the direction that we need to. Let’s look at the example below.
Example 1:
Given \(\triangle ROT\) has vertices \(R(2, 3), O(6, 1)\) and \(T(5, 4)\), rotate it \(90^{\circ}\) counterclockwise around the origin.
Step 2:
Since this is a \(90^{\circ}\) counter clockwise rotation the segment that we draw from the center of rotation will be perpendicular. This means that the slope of this segment will be the opposite reciprocal \(-\Large\frac{2}{3}\) and the distance will be the same from the center of rotation (up \(2\), left \(3\)). \(R'(-3, 2)\). |
Step 3:
The same approach is used to find the other points of the triangle. The slopes between \(O\) & the center is \(\Large\frac{4}{5}\) and between \(O'\) & the center is \(-\Large\frac{5}{4}\). The slopes between \(T\) & the center is \(\Large\frac{1}{6}\) and between \(T'\) and the center is \(-\Large\frac{6}{1}\). \(R' (-3, 2)\) \(O' (-4, 5)\) \(T’ (-1, 6)\) |
The same procedure of using slopes and distances can be applied for rotations of \(180^{\circ}\) and \(270^{\circ}\). Please practice these in the Quick Check problems.
Draw a Rotation in a Coordinate Plane
When a point is rotated counterclockwise about the origin, you can use the following rules:
When a point is rotated counterclockwise about the origin, you can use the following rules:
Practice a Rotation in a Coordinate Plane
When we rotate in coordinate plane, we use the following notation to indicate that which degree, direction and around which point we are rotating. For example: \(R_{\;-90^{\circ}}(2, -3)\) means that we are rotating \(90^{\circ}\) clockwise around the point \((2, -3)\).
Example 2:
\(\triangle ABC\) has vertices \(A(1, 2), B (4, 4)\) and \(C(5, 2)\). Graph its image after \(R_{90^{\circ}}(0, 0)\).
When we rotate in coordinate plane, we use the following notation to indicate that which degree, direction and around which point we are rotating. For example: \(R_{\;-90^{\circ}}(2, -3)\) means that we are rotating \(90^{\circ}\) clockwise around the point \((2, -3)\).
Example 2:
\(\triangle ABC\) has vertices \(A(1, 2), B (4, 4)\) and \(C(5, 2)\). Graph its image after \(R_{90^{\circ}}(0, 0)\).
Solution:
Example 3:
\(\triangle ABC\) has vertices \(A(1, 1), B (4, 4)\) and \(C(5, 2)\). Graph its image after \(R_{90^{\circ}}(-1, -1)\).
\(\triangle ABC\) has vertices \(A(1, 1), B (4, 4)\) and \(C(5, 2)\). Graph its image after \(R_{90^{\circ}}(-1, -1)\).
Solution:
Using slope and distance from each point to the center of rotation:
slope \(\overline{OA}=\Large\frac{2}{2}\), \(\perp\) slope = \(-\Large\frac{2}{2}\) (up \(2\), left \(2)\) slope \(\overline{OB}=\Large\frac{5}{4}\), \(\perp\) slope = \(-\Large\frac{4}{5}\) (up \(4\), left \(5\)) slope\(\overline{OC}=\Large\frac{3}{6}\), \(\perp\) slope = \(-\Large\frac{6}{3}\) (up \(6\), left \(3\)) \(A(1, 1)\rightarrow A'(-3, 1)\) \(B(4, 4)\rightarrow B'(-6, 3)\) \(C(5, 2)\rightarrow C'(-4, 5)\) |
Quick Check
1) Given Quadrilateral \(GAME\) has vertices \(G(2, 2), A(6, 1), M (5, 4)\) and \(E(2, 3)\).
a) rotate it \(180^{\circ}\) around the origin and graph the new figure \(G’A’M’E’\). Label its coordinates.
b) rotate it \(270^{\circ}\) counterclockwise around the origin and graph the new figure \(G’A’M’E’\). Label its coordinates.
1) Given Quadrilateral \(GAME\) has vertices \(G(2, 2), A(6, 1), M (5, 4)\) and \(E(2, 3)\).
a) rotate it \(180^{\circ}\) around the origin and graph the new figure \(G’A’M’E’\). Label its coordinates.
b) rotate it \(270^{\circ}\) counterclockwise around the origin and graph the new figure \(G’A’M’E’\). Label its coordinates.
2) Given \(\triangle BUS\) has vertices \(B(1, 3), U(2, 5)\) and \(S(4, 2)\) rotate it \(90^{\circ}\) counterclockwise around the origin and graph \(\triangle B’U’S’\). Label its coordinates.