Dilations Keep Lines Parallel - Geometry

1. Find the image of $y=-3x+9$ under dilation centered at $O=(0,0)$ with scale factor $k=2$.

   $y=-3x+18$. The $y$-intercept becomes $k \cdot 9 = 2 \cdot 9 = 18$.

2. A dilation centered at $O=(0,0)$ sends $y=mx+b$ to what line (assume $b\neq 0$)?

   $y=mx+kb$. The slope stays $m$, the $y$-intercept becomes $kb$.

3. Find the image of $y=3x+6$ under dilation centered at $O=(0,0)$ with scale factor $k=\frac{1}{2}$.

   $y=3x+3$. The $y$-intercept becomes $k \cdot 6 = \frac{1}{2} \cdot 6 = 3$.

4. Which option is correct for dilation: if $\ell \parallel m$, then $\ell' \parallel m'$?

   Yes, dilation preserves parallel lines: $\ell' \parallel m'$. Dilations map parallel lines to parallel lines.

5. Identify whether the image of $y=4x+1$ under dilation centered at $O=(0,0)$ is parallel or unchanged.

   Parallel (the line does not pass through $O$). The $y$-intercept is $1 \neq 0$, so line doesn't pass through $O$.

6. Which option correctly describes dilation of lines: preserves slope or changes slope unpredictably?

   It preserves slope for any line image. Both original and image lines have the same slope.

7. A dilation is centered at $O=(0,0)$. Identify the image of the line $x=0$ under $k=-4$.

   The same line $x=0$. The $y$-axis passes through the center of dilation.

8. A dilation is centered at $O=(0,0)$. Identify the image of the line $y=0$ under $k=\frac{3}{2}$.

   The same line $y=0$. The $x$-axis passes through the center of dilation.

9. Find the image of $y=-5$ under dilation centered at $O=(0,0)$ with scale factor $k=-2$.

   $y=10$. Apply the formula: $y=kc$ where $k=-2$ and $c=-5$.

10. Identify the correct conclusion: if a line maps to a distinct parallel line, what must be true about the center $O$?

    The original line does not pass through $O$. Lines not through the center map to distinct parallel lines.

11. What is the image of a segment lying on a line through $O$ under a dilation with scale factor $k$?

    A segment on the same line through $O$. The segment lies on the unchanged line through $O$.

12. What happens to the slope of a line under a dilation when the line does not pass through $O$?

    The slope is unchanged (the image line is parallel). Parallel lines have the same slope.

13. Find the image of $x=-4$ under dilation centered at $O=(0,0)$ with scale factor $k=\frac{1}{2}$.

    $x=-2$. Apply the formula: $x=kc$ where $k=\frac{1}{2}$ and $c=-4$.

14. Which condition guarantees that a dilation maps a line to a parallel line: the line passes through $O$ or not?

    The line must not pass through $O$. Lines not through the center map to parallel lines.

15. Which condition guarantees that a dilation maps a line to itself: the line passes through $O$ or not?

    The line must pass through $O$. Only lines through the center are unchanged by dilation.

16. What is true about angles formed by two lines after a dilation (assume both lines are not through $O$)?

    The angle measure is preserved. Parallel lines form angles with the same measure.

17. A dilation is centered at $O=(0,0)$. Identify the image of the line $y=-3x$ under $k=5$.

    The same line $y=-3x$. This line passes through the origin $(0,0)$.

18. Identify the correct conclusion: if a line is unchanged by a dilation, what must be true about the center $O$?

    The line must pass through $O$. Only lines through the center remain unchanged.

19. Identify the image of the line $y=3x$ under dilation centered at $O=(0,0)$ with scale factor $k=-1$.

    The same line $y=3x$. This line passes through the origin $(0,0)$.

20. Under dilation centered at $O=(0,0)$, what happens to the slope $m$ in $y=mx+b$?

    It stays $m$. Slope is preserved in the parallel image line.

21. Find the image of $y=2$ under dilation centered at $O=(0,0)$ with scale factor $k=3$.

    $y=6$. Apply the formula: $y=kc$ where $k=3$ and $c=2$.

22. Find the image of $y=-4x+\frac{3}{2}$ under dilation centered at $O=(0,0)$ with scale factor $k=2$.

    $y=-4x+3$. The $y$-intercept becomes $k \cdot \frac{3}{2} = 2 \cdot \frac{3}{2} = 3$.

23. Find the image of $y=x+2$ under dilation centered at $O=(0,0)$ with scale factor $k=4$.

    $y=x+8$. The $y$-intercept becomes $k \cdot 2 = 4 \cdot 2 = 8$.

24. Identify the invariant set for a dilation with center $O$: which lines are fixed pointwise as sets?

    All lines that pass through $O$ are fixed as sets. Lines through the center are the only invariant sets.

25. Find and correct the false claim: "If a line passes through $O$, a dilation makes a parallel line."

    Correct: If it passes through $O$, the line is unchanged. Lines through the center are unchanged, not made parallel.

26. A dilation has center $O$. Identify the image of the ray $\overrightarrow{OP}$.

    The same ray $\overrightarrow{OP}$. Rays through the center are unchanged by dilation.

27. Find the image of $y=\frac{1}{2}x-6$ under dilation centered at $O=(0,0)$ with scale factor $k=\frac{1}{3}$.

    $y=\frac{1}{2}x-2$. The $y$-intercept becomes $k \cdot (-6) = \frac{1}{3} \cdot (-6) = -2$.

28. A dilation has center $O$. Identify the image of a line $\ell$ that is parallel to $\overleftrightarrow{OP}$ but not through $O$.

    A line parallel to $\ell$. Parallel lines not through the center map to parallel lines.

29. What is the image of a line $\ell$ under a dilation with scale factor $k=1$?

    The same line $\ell$ (no change). Scale factor $k=1$ is the identity transformation.

30. A dilation has center $O$. Identify the image of the line through points $O$ and $P$.

    The same line $\overleftrightarrow{OP}$. Lines through the center are unchanged by dilation.

31. State the line-mapping rule for dilation: lines through $O$ and lines not through $O$.

    Through $O$: unchanged; not through $O$: mapped to a parallel line. This is the fundamental line-mapping property of dilation.

32. What is true about the intersection point of two lines through $O$ after a dilation?

    They still intersect at $O$ (the same point). Both lines pass through $O$ and remain unchanged.

33. What is the image of a line $\ell$ under a dilation with scale factor $k\neq 1$ if $\ell$ passes through $O$?

    The same line $\ell$ (still unchanged). Lines through the center are always unchanged.

34. A dilation has center $O$. Identify the image of a line segment $\overline{OP}$.

    A segment on the same line through $O$. The segment lies on the unchanged line through $O$.

35. Identify whether the image of $y=-2x$ under dilation centered at $O=(0,0)$ is parallel or unchanged.

    Unchanged (the line passes through $O$). This line passes through the origin $(0,0)$.

36. Find the image of $y=-x-8$ under dilation centered at $O=(0,0)$ with scale factor $k=\frac{1}{4}$.

    $y=-x-2$. The $y$-intercept becomes $k \cdot (-8) = \frac{1}{4} \cdot (-8) = -2$.

37. Identify the image of $y=5x$ under dilation centered at $O=(0,0)$ with scale factor $k=\frac{7}{3}$.

    The same line $y=5x$. This line passes through the origin $(0,0)$.

38. Find the image of $y=7x-\frac{9}{2}$ under dilation centered at $O=(0,0)$ with scale factor $k=-2$.

    $y=7x+9$. The $y$-intercept becomes $k \cdot (-\frac{9}{2}) = (-2) \cdot (-\frac{9}{2}) = 9$.

39. A dilation is centered at $O=(0,0)$. Identify the image of the line $y=mx$ under any scale factor $k$.

    The same line $y=mx$. Lines through the origin are unchanged by dilation.

40. What happens to the slope of a line under a dilation when the line passes through $O$?

    The slope is unchanged (the line is the same). The same line has the same slope.

41. Find the image of $x=7$ under dilation centered at $O=(0,0)$ with scale factor $k=-3$.

    $x=-21$. Apply the formula: $x=kc$ where $k=-3$ and $c=7$.

42. Which option is correct: a dilation can turn a line into a curve, or it always maps a line to a line?

    It always maps a line to a line. Dilations are linear transformations that preserve linearity.

43. What is the image of a segment lying on a line not through $O$ under a dilation with scale factor $k$?

    A segment on a line parallel to the original line. The original line maps to a parallel line.

44. Find and correct the false claim: "If a line does not pass through $O$, a dilation leaves it unchanged."

    Correct: It maps to a parallel line (usually distinct). Lines not through the center map to parallel lines.

45. What is the image of a horizontal line $y=c$ under a dilation centered at $O=(0,0)$ with factor $k$?

    The line $y=kc$ (parallel to $y=c$). Horizontal lines not through origin map to $y=kc$.

46. Identify the image of $y=0x-4$ under dilation centered at $O=(0,0)$ with scale factor $k=3$.

    $y=-12$. The $y$-intercept becomes $k \cdot (-4) = 3 \cdot (-4) = -12$.

47. Identify the image of the line $y=-9$ under dilation centered at $O=(0,0)$ with scale factor $k=-\frac{1}{3}$.

    $y=3$. The $y$-intercept becomes $k \cdot (-9) = (-\frac{1}{3}) \cdot (-9) = 3$.

48. Identify the image of $x=2$ under dilation centered at $O=(0,0)$ with scale factor $k=\frac{5}{2}$.

    $x=5$. Apply the formula: $x = k \cdot 2 = \frac{5}{2} \cdot 2 = 5$.

49. Under dilation centered at $O=(0,0)$, what happens to the $y$-intercept $b$ in $y=mx+b$?

    It becomes $kb$. The $y$-intercept is multiplied by the scale factor $k$.

50. Identify the correct statement: A dilation preserves line parallelism or makes all lines intersect?

    A dilation preserves parallelism of lines. Dilations preserve parallel relationships between lines.

51. What is the image of a line $\ell$ under a dilation if $\ell$ passes through the dilation center $O$?

    The same line $\ell$ (it is unchanged). Lines through the center remain fixed as sets.

52. What is the image of a vertical line $x=c$ under a dilation centered at $O=(0,0)$ with factor $k$?

    The line $x=kc$ (parallel to $x=c$). Vertical lines not through origin map to $x=kc$.

53. What is the image of a line $\ell$ under a dilation if $\ell$ does not pass through the dilation center $O$?

    A line $\ell'$ parallel to $\ell$. Lines not through the center map to parallel lines.

54. Find the image of $y=2x-10$ under dilation centered at $O=(0,0)$ with scale factor $k=-\frac{1}{2}$.

    $y=2x+5$. The $y$-intercept becomes $k \cdot (-10) = -\frac{1}{2} \cdot (-10) = 5$.

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