- Figure ABCD is reflected across line m to create figure A'B'C'D'. Then figure A'B'C'D' is reflected across line n, where line n is parallel to line m and 5 units away from it. What single transformation is equivalent to these two reflections?
Answer: A translation of 10 units in the direction perpendicular to both lines
When a figure is reflected across two parallel lines, the result is equivalent to a translation. The distance of translation is twice the distance between the parallel lines, in the direction from the first line to the second line. Since the lines are 5 units apart, the translation distance is 2 × 5 = 10 units. Choice A describes reflection across perpendicular lines, Choice C uses the wrong distance, Choice D would result in the original figure.
- A square with side length 4 is rotated 45° clockwise about its center, then dilated by scale factor 21 about the same center point. What is the area of the final image compared to a square with the same orientation as the original that circumscribes the final image?
Answer: Final image area: 4; circumscribing square area: 8
Original square has area 16. After rotation, area stays 16 but orientation changes (becomes diamond-shaped relative to coordinate axes). After dilation by 1/2, area becomes 16 × (1/2)² = 4. The rotated square with side length 2 fits inside a square aligned with axes. The diagonal of the rotated square is 2√2, so the circumscribing axis-aligned square has side length 2√2 and area (2√2)² = 8. Choice B uses original area, Choice C uses pre-dilation area, Choice D has wrong calculation.
- A triangle is dilated by a scale factor of 3 with center at point C, which is one of the triangle's vertices. After dilation, the triangle is rotated 45° clockwise about the same point C. Which property of the triangle is preserved throughout both transformations?
Answer: The measure of all three angles and the position of vertex C
Dilation preserves angle measures but changes side lengths by the scale factor (3). Since C is the center of dilation, point C doesn't move. Rotation preserves all distances and angles, and since C is the center of rotation, C remains fixed. Therefore, all angle measures and the position of vertex C are preserved. Choice B is wrong because dilation changes side lengths, Choice C is wrong because dilation changes area by factor of 9, Choice D is wrong because dilation changes perimeter by factor of 3.
- A regular hexagon centered at the origin undergoes a dilation with scale factor 2, followed by a reflection across the x-axis. If a vertex of the original hexagon was at (3, 0), which statement correctly describes the combined transformation?
Answer: The final image has a vertex at (6, 0) and the hexagon's orientation is reversed
First, dilation by scale factor 2: vertex (3, 0) → (6, 0). Then reflection across x-axis: (6, 0) → (6, 0) since it's on the x-axis. However, other vertices will show the orientation change. A reflection reverses orientation (clockwise ↔ counterclockwise). Choice A is incorrect as this isn't equivalent to reflection across y = -x, Choice C has wrong coordinates, Choice D incorrectly states orientation is maintained.
