Dilations Change Length by Scale Factor - Geometry

1. What is the scale factor $k$ if a segment is halved in length by a dilation?

   $k=\frac{1}{2}$. Halving means multiplying by $\frac{1}{2}$.

2. What is the image length $L'$ if $L=\frac{3}{5}$ and the scale factor is $k=\frac{10}{3}$?

   $L'=2$. Using $L'=kL$: $L'=\frac{3}{5} \times \frac{10}{3}=2$.

3. Identify whether the dilation is an enlargement or reduction if the scale factor is $k=\frac{11}{10}$.

   Enlargement. Since $k=\frac{11}{10}>1$, it's an enlargement.

4. Identify the correct inequality if $k=\frac{5}{6}$ and $L>0$ for a dilated segment length $L'$.

   $L'<L$. Since $k=\frac{5}{6}<1$, the image is shorter.

5. What is the scale factor $k$ if a dilation decreases every length by $10\%$?

   $k=0.9$. 10% decrease means keeping 90%, so $k=0.9$.

6. What is the scale factor $k$ if $L=16$ and $L'=10$ under a dilation?

   $k=\frac{5}{8}$. Scale factor is $k=\frac{10}{16}=\frac{5}{8}$.

7. What is the image length $L'$ if $L=11$ and the scale factor is $k=1.2$?

   $L'=13.2$. Using $L'=kL$: $L'=11 \times 1.2=13.2$.

8. What is the image length $L'$ if $L=\frac{5}{2}$ and the scale factor is $k=4$?

   $L'=10$. Using $L'=kL$: $L'=\frac{5}{2} \times 4=10$.

9. What is the image length $L'$ if $L=15$ and the scale factor is $k=0.6$?

   $L'=9$. Using $L'=kL$: $L'=15 \times 0.6=9$.

10. What is the original length $L$ if $L'=6$ and the scale factor is $k=\frac{3}{2}$?

    $L=4$. From $L'=kL$: $L=\frac{6}{\frac{3}{2}}=4$.

11. What is the scale factor $k$ if a segment changes from $12$ to $18$ under dilation?

    $k=\frac{3}{2}$. Scale factor is $k=\frac{18}{12}=\frac{3}{2}$.

12. Find the missing value: if $L'=kL$, $L=12$, and $L'=15$, what is $k$?

    $k=\frac{5}{4}$. From $L'=kL$: $k=\frac{15}{12}=\frac{5}{4}$.

13. What does a scale factor $k=0$ do to any segment length under dilation about a center?

    It collapses to $0$; $L'=0$. Scale factor 0 makes all segments disappear to a point.

14. What is the image length $L'$ if $L=48$ and the dilation decreases lengths by $25\%$?

    $L'=36$. 25% decrease means $k=0.75$, so $L'=48 \times 0.75=36$.

15. What is the scale factor $k$ if $L=7$ and $L'=\frac{21}{2}$?

    $k=\frac{3}{2}$. Scale factor is $k=\frac{\frac{21}{2}}{7}=\frac{3}{2}$.

16. What is the scale factor $k$ if a segment changes from $30$ to $24$ under dilation?

    $k=0.8$. Scale factor is $k=\frac{24}{30}=0.8$.

17. What does a scale factor $k>1$ do to the length of a line segment under dilation?

    It increases length; $L'>L$. Scale factors greater than 1 stretch segments longer.

18. What is the ratio $\frac{L'}{L}$ when the scale factor is $k=0.75$?

    $\frac{L'}{L}=0.75$. The ratio of image to original equals the scale factor.

19. What is the ratio of preimage length to image length for a dilation with scale factor $k$ (and $k\ne 0$)?

    $\frac{L}{L'}=\frac{1}{k}$. The reciprocal relationship when dividing original by image.

20. Find the missing value: if $L'=kL$, $k=\frac{2}{5}$, and $L'=8$, what is $L$?

    $L=20$. From $L'=kL$: $L=\frac{8}{\frac{2}{5}}=20$.

21. What is the scale factor $k$ if a segment of length $L$ dilates to length $L'$?

    $k=\frac{L'}{L}$. Rearranging $L'=kL$ to solve for the scale factor.

22. What is the original length $L$ if $L'=2.5$ and the scale factor is $k=0.5$?

    $L=5$. From $L' = k L$: $L=\frac{2.5}{0.5}=5$.

23. Find and correct the error: "Under dilation, $L'=L+k$." What is the correct relationship?

    Correct: $L'=kL$. Dilation multiplies length, not adds to it.

24. What is the image length $L'$ if $L=13$ and the scale factor is $k=2$?

    $L'=26$. Using $L'=kL$: $L'=13 \times 2=26$.

25. What is the original length $L$ if $L'=27$ and the dilation scale factor is $k=\frac{3}{2}$?

    $L=18$. From $L'=kL$: $L=\frac{27}{\frac{3}{2}}=18$.

26. Identify the scale factor $k$ if the statement "image length : preimage length $=3:5$" is true.

    $k=\frac{3}{5}$. The ratio $3:5$ means $k=\frac{3}{5}$.

27. What is the image length $L'$ if $L=18$ and the dilation increases lengths by $25\%$?

    $L'=22.5$. 25% increase means $k=1.25$, so $L'=18 \times 1.25=22.5$.

28. What is the scale factor $k$ if $L=25$ and $L'=30$ under dilation?

    $k=1.2$. Scale factor is $k=\frac{30}{25}=1.2$.

29. What is the image length $L'$ if $L=9$ and the scale factor is $k=\frac{2}{3}$?

    $L'=6$. Using $L'=kL$: $L'=9 \times \frac{2}{3}=6$.

30. What is the image length $L'$ if $L=8$ and the scale factor is $k=\frac{3}{2}$?

    $L'=12$. Using $L'=kL$: $L'=8 \times \frac{3}{2}=12$.

31. Identify whether the dilation is an enlargement or reduction if the scale factor is $k=\frac{9}{10}$.

    Reduction. Since $k=\frac{9}{10}<1$, it's a reduction.

32. What is the ratio $\frac{L'}{L}$ when the scale factor is $k=\frac{9}{4}$?

    $\frac{L'}{L}=\frac{9}{4}$. The ratio of image to original equals the scale factor.

33. What is the scale factor $k$ if a segment is reduced to one fifth of its original length?

    $k=\frac{1}{5}$. One fifth means multiply by $\frac{1}{5}$.

34. What is the image length $L'$ if $L=13$ and the scale factor is $k=\frac{1}{2}$?

    $L'=6.5$. Using $L'=kL$: $L'=13 \times \frac{1}{2}=6.5$.

35. Identify the correct inequality if $k=\frac{7}{4}$ and $L>0$ for a dilated segment length $L'$.

    $L'>L$. Since $k=\frac{7}{4}>1$, the image is longer.

36. What is the image length $L'$ if $L=20$ and the scale factor is $k=\frac{1}{4}$?

    $L'=5$. Using $L'=kL$: $L'=20 \times \frac{1}{4}=5$.

37. What is the image length $L'$ if $L=50$ and the scale factor is $k=0.02$?

    $L'=1$. Using $L'=kL$: $L'=50 \times 0.02=1$.

38. What is the formula for the image length $L'$ after a dilation with scale factor $k$ from length $L$?

    $L'=kL$. The dilation formula multiplies original length by the scale factor.

39. Find and correct the error: "Scale factor $k=\frac{L}{L'}$." What is the correct formula for $k$?

    Correct: $k=\frac{L'}{L}$. Scale factor is image divided by original length.

40. What is the scale factor $k$ if a dilation increases every length by $10\%$?

    $k=1.1$. 10% increase means 110% of original, so $k=1.1$.

41. What is the scale factor $k$ if a segment changes from $\frac{9}{2}$ to $3$ under dilation?

    $k=\frac{2}{3}$. Scale factor is $k=\frac{3}{\frac{9}{2}}=\frac{2}{3}$.

42. What is the image length $L'$ if $L=36$ and the dilation scale factor is $k=\frac{5}{12}$?

    $L'=15$. Using $L'=kL$: $L'=36 \times \frac{5}{12}=15$.

43. What does a scale factor $0<k<1$ do to the length of a line segment under dilation?

    It decreases length; $L'<L$. Scale factors between 0 and 1 shrink segments shorter.

44. What is the length scale factor if the dilation is described as "reduction to $40\%$ of original"?

    $k=0.4$. 40% of original means multiply by 0.4.

45. What is the scale factor $k$ if a segment is tripled in length by a dilation?

    $k=3$. Tripling means multiplying by 3.

46. What is the original length $L$ if the image length is $L'=9$ and $k=\frac{3}{4}$?

    $L=12$. From $L'=kL$: $L=\frac{9}{\frac{3}{4}}=12$.

47. What is the ratio of image length to preimage length for a dilation with scale factor $k$?

    $\frac{L'}{L}=k$. The ratio of lengths equals the scale factor.

48. What is the original length $L$ if the image length is $L'=14$ and $k=2$?

    $L=7$. From $L'=kL$: $L=\frac{L'}{k}=\frac{14}{2}=7$.

49. What is the scale factor $k$ if a segment of length $0.8$ dilates to length $2.4$?

    $k=3$. Scale factor is $k=\frac{2.4}{0.8}=3$.

50. Identify the scale factor $k$ if the statement "preimage length : image length $=4:7$" is true.

    $k=\frac{7}{4}$. Ratio $4:7$ (original:image) means $k=\frac{7}{4}$.

51. What is the ratio of preimage length to image length for a dilation with scale factor $k$ (and $k \neq 0$)?

    $\frac{L}{L'}=\frac{1}{k}$. The reciprocal relationship when dividing original by image.

52. What is the length scale factor if the dilation is described as "enlargement by $150\%$"?

    $k=1.5$. 150% means 1.5 times the original size.

53. What does a scale factor $k=1$ do to the length of a line segment under dilation?

    It keeps the same length; $L'=L$. Scale factor 1 produces no change in length.

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