Rotations and Rotational Symmetry

# Rotations and Rotational Symmetry - Date'2’l4“ l"E...

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Unformatted text preview: Date: '2’l4“ l";E Q7 Transformations — Rotations and Rotational Symmetry - Notes Day 3 r0 honrsatransdrma onmw c a gureis aroun apom c point of rotation. The image of a rotated ﬁgure has all the same angle measures and lengths as the original and differs only in position. Rotations that are counter— clockwise are rotations of positive degree measm'e. Rotations that are clock-wise have a negative degree measure. The two rotated right triangles illustrate counter- clockwise positive 90° rotation and clockwise —90° rotatiori. counterclockwise ‘POSVYNQ A ﬁgure has rotational symmetry if the ﬁgure is its own image under a rotation of more than 0" but less than 60°. Rotational symmetry of exactly 180° is the same as point symmetry or a dilation of ~1. Using letters of the alphabet with point synuhetry (Z, X, S, O, N, l, H), it is easy to see—by turning the page upside down—that they have rotational symmetry. 0 However, the converse is not true. For example, equilateral triangle ABC has rota- tional symmetry of 120°, but it does not have point symmetry. B A ‘ c . i.» <3 no as no Many letters, as well as designs in the H S N , Any regular polygon has rotational symmetry. When regular pentagon ABCDE is rotated 1:2, or 72", about its shapes of wheels, stars, and polygons, 6 «m a have rotational symmetry. Each ﬁgure {610 I 6 _ shown at the right has rotariorial sym- {lg merry. r B | . center, the image of every point of the ﬁgure is a point of the ﬁgure. Under this rotation, A -> B, B —> C, C —> D, ' D—>E,andE-%A. The ﬁgure would also have rotational symmetry if rotated through a multiple of 72° (144°, 216“, or 288°). If it were rotated through 360°, every point would be its own image. Since this is true for every ﬁgure, we do not usually consider a 360° rorarion as rotational symmetry. 7 Rotations A rotation can be thought of as turning a figure about some fixed point. The ' ' ‘ ' ’ t P so an 1e of 180° counter— clockwise): Examples: If equilateral triangle ABC is rotated 1808 about point P the result (image) will be equilateral triangle A'B'C‘. This is also called a half-turn. 1. Figure B is the image of figure A ‘ 3. If T is rotated 90”, the image is under which transformation? ' (l) -l (3) J. i“? r— (a) T (1) line reflection ’ rotation 3 translation } dilation If K,is rotated 180°, which is the resulting figure? In the accompanying diagram, a (1) g )l square is inscribed in circle 0. Find the result of each of the (2) 7‘ (A) K following rotations. For each of the following letters draw the resulting figure under the rotation given. {2) 9" Rmo" H: R —QO‘J _——1_i L i Q u: R270” (5) YI, R1300 3(00 5- Which ﬁgure has 36° rotational symmetry? . W {1) regular pentagon (3) reguiar octagon @ regular decagon (4) ' reguiar dodecagon 2- che1etter P is rotated 180 degrees, which is the . resulting ﬁgure? I A (9 d (2):). (3)"5 (4)!) 3. Which symbol has 90° rotational symmetry? @X (2)H (3)5 (4)8 - LL Which ﬁgure has 60° rotational symmetry? 34.6%..) (1) square (3 regular octagon (2) equilateral triangle regular. hexagon 5. Which letter has int symmetry? (1) A (2) B N (4) T (I) E (2) T (3) C 7. Which ﬁgure has 180" rotational symmetry? éﬁnﬁ 3' Name the image of point A under 'a clockwise rote- tion with center 0 of: a. 60" E: d. 300° 1). 360° 8. 240° c. 130‘ f. 120° 9- The vertices of AABC are A(1, 3), 3(5, 1), and 0(1, 1). On one set of axes, draw AABC and its image AA'B'C' under a rotation of 90° coun- terclockwise about the origin. ‘ gecgn / P)‘: L“1 33) -.-r / ‘ZGZJD ...
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