Rotations

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.

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.

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).

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′.

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).

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.

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).

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)

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’.

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′.

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)?

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.

Page 513

Lesson Practice (Ask Mr. Heintz)

Page 513

Practice 1-30 (Do the starred ones first)