rotations -...
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![Page 1: Rotations - bfhscooney.weebly.combfhscooney.weebly.com/uploads/5/4/4/3/54438819/lesson_78-geom… · Rotations - A rotation is an isometry, meaning that the preimage and its rotated](https://reader034.vdocument.in/reader034/viewer/2022050519/5fa313ad7135752be91a47af/html5/thumbnails/1.jpg)
Rotations
![Page 2: Rotations - bfhscooney.weebly.combfhscooney.weebly.com/uploads/5/4/4/3/54438819/lesson_78-geom… · Rotations - A rotation is an isometry, meaning that the preimage and its rotated](https://reader034.vdocument.in/reader034/viewer/2022050519/5fa313ad7135752be91a47af/html5/thumbnails/2.jpg)
A rotation is a transformation about a point. The center of rotation is the fixed point around which a figure is rotated. Rotations are assumed to be measured as a counterclockwise turn unless otherwise stated.
![Page 3: Rotations - bfhscooney.weebly.combfhscooney.weebly.com/uploads/5/4/4/3/54438819/lesson_78-geom… · Rotations - A rotation is an isometry, meaning that the preimage and its rotated](https://reader034.vdocument.in/reader034/viewer/2022050519/5fa313ad7135752be91a47af/html5/thumbnails/3.jpg)
Rotations - A rotation is an isometry, meaning that the preimage and its rotated image are the same shape and size.
In the diagram, ∆𝐴𝐵𝐶 has been rotated around the center of rotation R. The resulting image, ∆𝐴′𝐵′𝐶′, is congruent to ∆𝐴𝐵𝐶.
![Page 4: Rotations - bfhscooney.weebly.combfhscooney.weebly.com/uploads/5/4/4/3/54438819/lesson_78-geom… · Rotations - A rotation is an isometry, meaning that the preimage and its rotated](https://reader034.vdocument.in/reader034/viewer/2022050519/5fa313ad7135752be91a47af/html5/thumbnails/4.jpg)
For rotations around the origin with angles of rotation that are multiples of 90°, the following transformation mapping notations apply. If a point (x, y) is rotated 90° about the origin:
T: (x, y) → (-y, x).
If a point (x, y) is rotated 180° about the origin: T: (x, y) → (-x, -y).
If a point (x, y) is rotated 270° about the origin: T: (x, y) → (y, -x).
![Page 5: Rotations - bfhscooney.weebly.combfhscooney.weebly.com/uploads/5/4/4/3/54438819/lesson_78-geom… · Rotations - A rotation is an isometry, meaning that the preimage and its rotated](https://reader034.vdocument.in/reader034/viewer/2022050519/5fa313ad7135752be91a47af/html5/thumbnails/5.jpg)
If ∆𝑀𝑁𝑃 has vertices M(1, 1), N(4, 3), and P(5, 2), graph the triangle and its rotation 180° counterclockwise about the origin. SOLUTION Graph ∆𝑀𝑁𝑃 on a coordinate plane. Use the rule for a 180° rotation to find the vertices of the rotated triangle, ∆𝑀′𝑁′𝑃′. T: (x, y) → (-x, -y) M(1, 1) → M’(-1, -1) N(4, 3) → N’(-4, -3) P(5, 2) → P’(-5, -2) Graph ∆𝑀′𝑁′𝑃′.
![Page 6: Rotations - bfhscooney.weebly.combfhscooney.weebly.com/uploads/5/4/4/3/54438819/lesson_78-geom… · Rotations - A rotation is an isometry, meaning that the preimage and its rotated](https://reader034.vdocument.in/reader034/viewer/2022050519/5fa313ad7135752be91a47af/html5/thumbnails/6.jpg)
Rotating a figure around a point that is not the origin can be more difficult.
For a 180° rotation, there is still a simple transformation mapping, given below.
If a point (x, y) is rotated 180° about the point (a, b):
T: (x, y) → (2a - x, 2b - y).
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To rotate a figure 90° around a point that is not the origin, consider the diagram of the point E. To rotate E around the point (a, b), notice that the two points have the same y-coordinate. After a 90° turn, the new point, F, will lie directly above the point of rotation and therefore will have the same x-coordinate. It will remain the same distance away from the point of rotation. If, for example, E was originally 2 units to the right of the point of rotation, F will be 2 units above the point of rotation.
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a. Rotate the point (-3, 4) 90° counterclockwise about the point (-3, 6). SOLUTION Plot the points as shown in the diagram. After a 90° rotation, the rotated image of (-3, 4) will lie to the right of the center of rotation. The point (-3, 4), by application of the distance formula, is 2 units away from the center of rotation. The image will also be 2 units away, but will lay 2 units to the right instead of 2 units below the center of rotation. Count 2 units over from the center of rotation or add 2 to the center of rotation’s x-coordinate. The rotated point is (-1, 6).
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b. Rotate the point (7, 8) 180° around the center of rotation (–2, 3).
SOLUTION
Use the transformation mapping given above:
T: (x, y) → (2a - x, 2b - y).
T: (7, 8) → (2(-2) - 7, 2(3) - 8)
T: (7, 8) → (-11, -2)
![Page 10: Rotations - bfhscooney.weebly.combfhscooney.weebly.com/uploads/5/4/4/3/54438819/lesson_78-geom… · Rotations - A rotation is an isometry, meaning that the preimage and its rotated](https://reader034.vdocument.in/reader034/viewer/2022050519/5fa313ad7135752be91a47af/html5/thumbnails/10.jpg)
Look at the Ferris wheel at right. Find the angle of rotation that transforms M to move to M’. Explain. SOLUTION There are 20 supporting arms, or spokes, on the Ferris wheel. Find the measure of the angle made by two adjacent spokes. 360° ÷ 20 = 18° The spokes divide the wheel into 20 angles of 18°. M’ is 7 places counterclockwise from M. 18° × 7 = 126° The Ferris wheel must rotate 126° for a seat in position M to move to M’.
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a.∆𝐴𝐵𝐶 has vertices A(-2, -3), B(1, 1), and C(2, -1). Graph ∆𝐴𝐵𝐶 and its image after a 90° rotation. List the coordinates of the vertices of ∆𝐴′𝐵′𝐶′.
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b.Triangle DEF has vertices D(0, -2), E(1, 0), and F(3, -1). Find the coordinates of the vertices of the image if ∆𝐷𝐸𝐹 is rotated 180° about the point R(-1, 1)?
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c. Automotive Look at the rim of a car tire. To the nearest degree, find the angle of rotation required for support spoke A to move counterclockwise to position B. Explain.
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Lesson Practice (Ask Mr. Heintz)
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Practice 1-30 (Do the starred ones first)