Describing Transformations - Math 1

1. Triangle $ABC$ is mapped to triangle $A'B'C'$ by a transformation. Point $A(2, 3)$ maps to $A'(6, -1)$, point $B(5, 7)$ maps to $B'(9, 3)$, and point $C(1, 6)$ maps to $C'(5, 2)$. Which transformation best describes this mapping?
   Answer: A translation by the vector $\langle 4, -4 \rangle$ preserving orientation and distance To find the transformation, examine the change from each point to its image. For all three points: $A(2,3) \to A'(6,-1)$ gives vector $\langle 4, -4 \rangle$, $B(5,7) \to B'(9,3)$ gives vector $\langle 4, -4 \rangle$, and $C(1,6) \to C'(5,2)$ gives vector $\langle 4, -4 \rangle$. Since all points move by the same vector, this is a pure translation by $\langle 4, -4 \rangle$. Choice A is incorrect because no reflection is needed. Choice C is incorrect because rotation would change the orientation differently. Choice D is incorrect because reflection across $y = x$ would swap coordinates.
2. A regular octagon is rotated about its center such that it maps onto itself. If the smallest positive angle of rotation that achieves this mapping is applied, what fraction of a complete revolution does this represent?
   Answer: $\frac{1}{8}$ of a complete revolution, representing the fundamental rotational symmetry unit A regular octagon has 8-fold rotational symmetry, meaning it maps onto itself under rotations of $\frac{360°}{8} = 45°$. The smallest positive rotation that maps the octagon onto itself is $45°$, which represents $\frac{45°}{360°} = \frac{1}{8}$ of a complete revolution. Choice B corresponds to $90°$, which would work but is not the smallest angle. Choice C corresponds to $60°$, which does not map a regular octagon onto itself. Choice D corresponds to $22.5°$, which is too small to map the octagon onto itself.
3. A transformation $T$ maps parallelogram $WXYZ$ to parallelogram $W'X'Y'Z'$ such that the area is preserved, all angles remain congruent to their corresponding angles, but the orientation is reversed. Additionally, no point remains fixed under this transformation. Which type of transformation could $T$ represent?
   Answer: A reflection across a line that does not intersect the parallelogram, maintaining area and angles with orientation change The transformation preserves area and angles but reverses orientation with no fixed points. A reflection across a line that doesn't intersect the parallelogram satisfies all these conditions: it preserves area and angles (isometry), reverses orientation, and has no fixed points since the line doesn't touch the figure. Choice A (rotation) preserves orientation rather than reversing it. Choice C (translation + rotation) would preserve orientation if the rotation isn't 180°, and if it is 180° it would be equivalent to a reflection. Choice D (glide reflection) is possible but the specific distance condition is unnecessarily restrictive and not required by the given conditions.
4. An equilateral triangle with side length $6$ is inscribed in a circle. The triangle undergoes a rotation about the circle's center such that each vertex moves to the position of the next vertex in the clockwise direction. What is the precise description of this rotation?
   Answer: A clockwise rotation of $120°$ about the center preserving the triangular symmetry pattern When you encounter problems involving regular polygons inscribed in circles, focus on the rotational symmetries these shapes possess. An equilateral triangle has rotational symmetry, meaning it looks identical after certain rotations about its center. To find the rotation angle, consider that the triangle has 3 vertices equally spaced around the circle. Since a full rotation is $360°$, each vertex is separated by $360° ÷ 3 = 120°$. When each vertex moves to the next vertex's position clockwise, the triangle rotates $120°$ clockwise about the center. This rotation preserves the triangle's appearance because of its symmetrical properties. Choice A incorrectly states $240°$ counterclockwise. While $240°$ counterclockwise is equivalent to $120°$ clockwise, the problem specifically describes clockwise motion, making this description imprecise. Choice B suggests $120°$ counterclockwise, which would move each vertex to the previous vertex's position, opposite to what's described. Choice C claims $60°$ clockwise, but this angle corresponds to moving halfway between vertices, not to the next vertex position. Choice D correctly identifies the $120°$ clockwise rotation and accurately notes that this preserves the triangular symmetry pattern, since the triangle appears unchanged after this rotation. Study tip: For any regular $n$-sided polygon, the rotational symmetry angle is $360°/n$. Memorize this formula and remember that rotations preserving the shape's appearance are always multiples of this base angle.
5. A transformation maps every point $(x, y)$ to $(3-y, x+2)$. When this transformation is applied to a square with vertices at $(0, 0)$, $(2, 0)$, $(2, 2)$, and $(0, 2)$, which statement most accurately describes the result?
   Answer: The image is a congruent square rotated $90°$ counterclockwise with appropriate translation Applying $(x,y) \rightarrow (3-y, x+2)$ to vertices: $(0,0) \rightarrow (3,2)$, $(2,0) \rightarrow (3,4)$, $(2,2) \rightarrow (1,4)$, $(0,2) \rightarrow (1,2)$. The transformation preserves side lengths and angles, producing a congruent square. The mapping $(x,y) \rightarrow (-y,x)$ represents a 90° counterclockwise rotation, and the $(3,2)$ addition represents translation. Choice C correctly identifies this as a rotation followed by translation.
6. A transformation maps point $P(a, b)$ to point $P'(3a + 2, -b + 1)$. This transformation is applied to rectangle $DEFG$ with vertices $D(0, 0)$, $E(4, 0)$, $F(4, 3)$, and $G(0, 3)$. Which statement best describes the resulting figure?
   Answer: The image is a rectangle with area three times the original, with vertices in reversed orientation Applying the transformation: $D(0,0) \to D'(2,1)$, $E(4,0) \to E'(14,1)$, $F(4,3) \to F'(14,-2)$, $G(0,3) \to G'(2,-2)$. The transformation $(x,y) \to (3x+2, -y+1)$ includes a horizontal stretch by factor 3, a reflection across a horizontal line, and translations. The area scales by factor 3 (from the $x$-stretch), and the reflection reverses orientation. The image remains rectangular because the transformation preserves right angles. Choice A is wrong about area. Choice C is wrong about orientation. Choice D is wrong about both area and the specific reflection line.
7. A regular hexagon centered at the origin is rotated about its center. After the rotation, vertex $A$ coincides with the position where vertex $C$ was originally located. If vertices are labeled consecutively around the hexagon, what is the most complete description of this rotation?
   Answer: A counterclockwise rotation of $120°$ about the center preserving distance and angle measures In a regular hexagon with consecutively labeled vertices, vertex $A$ is separated from vertex $C$ by 2 positions out of 6 total vertices. Each vertex position corresponds to $\frac{360°}{6} = 60°$. Moving from $A$ to $C$ requires rotating 2 positions, which is $2 \times 60° = 120°$. Since $A$ moves to where $C$ was, this is a counterclockwise rotation of $120°$. Choice A gives the wrong direction. Choice B uses the wrong angle. Choice D is incorrect because $240°$ clockwise equals $120°$ clockwise, not counterclockwise.
8. A quadrilateral undergoes a transformation where each vertex $(x, y)$ is mapped to $(-y, x)$. The transformed figure is then reflected across the $y$-axis. Which single transformation is equivalent to this composition?
   Answer: A rotation of $90°$ clockwise about the origin maintaining distance properties The transformation $(x,y) \to (-y,x)$ is a $90°$ counterclockwise rotation. Then reflecting across the $y$-axis maps $(-y,x) \to (y,x)$. The composition maps $(x,y) \to (y,x)$, which is equivalent to a $90°$ clockwise rotation about the origin. Choice A describes the two-step process but doesn't identify the equivalent single transformation. Choice B is incorrect because reflection across $y = -x$ maps $(x,y) \to (-y,-x)$. Choice D is incorrect because this composition would map $(x,y) \to (-x,y)$.
9. Line segment $\overline{MN}$ with endpoints $M(-3, 4)$ and $N(5, -2)$ is transformed such that $M$ maps to $M'(4, 3)$ and $N$ maps to $N'(-2, -5)$. What type of transformation preserves the length of the segment while achieving this mapping?
   Answer: A reflection across the line $y = -x + 1$ preserving perpendicular distance relationships First, verify that segment length is preserved: $|\overline{MN}| = \sqrt{(5-(-3))^2 + (-2-4)^2} = \sqrt{64 + 36} = 10$ and $|\overline{M'N'}| = \sqrt{(-2-4)^2 + (-5-3)^2} = \sqrt{36 + 64} = 10$. The length is preserved. To find the transformation, note that the midpoint of $\overline{MN}$ is $(1, 1)$ and the midpoint of $\overline{M'N'}$ is $(1, -1)$. The perpendicular bisector of the segment connecting these midpoints is $y = -x + 1$. Reflection across this line maps $M$ to $M'$ and $N$ to $N'$. Choices B, C, and D do not produce the correct image points when applied to the original coordinates.

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