Circle Similarity - Geometry

1. What is the circle equation after dilating $x^2+y^2=16$ about the origin by $k = \frac{1}{2}$?

   $x^2 + y^2 = 4$. Radius scales from $4$ to $2$: $(\frac{1}{2})^2 \times 16 = 4$.

2. What is the radius after dilating a circle of radius $5$ by $k = -2$?

   $10$. Take absolute value: $|{-2}| \times 5 = 10$.

3. What is the center of the circle $(x-8)^2+(y+3)^2=16$?

   $(8,-3)$. Center coordinates are $(8, -3)$ from the standard form.

4. What transformation type is sufficient to prove two circles are similar?

   A dilation (possibly followed by a rigid motion). Similarity requires at most one dilation plus rigid motions.

5. Identify the similarity transformation that maps $x^2 + y^2 = 1$ to $x^2 + y^2 = 9$.

   A dilation about the origin with $k = 3$. Radius increases from $1$ to $3$, so scale factor is $3$.

6. What is the image of a circle under a translation by vector $\langle a,b \rangle$?

   A congruent circle with its center shifted by $\langle a,b \rangle$. Translation preserves size and shape, only changes position.

7. What is a correct similarity mapping from the unit circle to a circle of radius $R$?

   A dilation by $k = R$ (about the same center). Scale up by target radius from unit circle.

8. Identify the scale factor mapping $x^2+y^2=25$ to $x^2+y^2=1$.

   $k = \frac{1}{5}$. Scale from radius $5$ down to radius $1$.

9. What is the scale factor that maps radius $12$ to radius $3$?

   $k = \frac{3}{12} = \frac{1}{4}$. Scale factor is target divided by source: $\frac{3}{12}$.

10. What is the center after translating a circle centered at $(2,-1)$ by $\langle 5,4 \rangle$?

    $(7,3)$. Add translation vector components: $(2+5, -1+4)$.

11. What is the radius after dilating a circle of radius $7$ by $k = \frac{3}{2}$?

    $\frac{21}{2}$. Multiply original radius by scale factor: $7 \times \frac{3}{2}$.

12. What is the preimage radius if a dilation with $k = \frac{2}{5}$ produces radius $8$?

    $20$. If $\frac{2}{5}r = 8$, then $r = 8 \times \frac{5}{2} = 20$.

13. What is the central angle measure preserved by a dilation between two circles?

    The corresponding central angles are equal in measure. Dilation preserves angles, so central angles remain unchanged.

14. What is a correct similarity mapping from a circle of radius $r$ to a unit circle?

    A dilation by $k = \frac{1}{r}$ (about the same center). Scale down by reciprocal of radius to get unit circle.

15. What is the scale factor if similar circles have diameters $d_1$ and $d_2$?

    $k = \frac{d_2}{d_1}$. Diameter and radius have the same scale factor.

16. What is the scale factor mapping radius $r_1$ to radius $r_2$ when $r_1=r_2$?

    $k = 1$. Equal radii means no scaling needed.

17. What is the ratio of circumferences of similar circles with radii $r_1$ and $r_2$?

    $\frac{C_2}{C_1} = \frac{r_2}{r_1}$. Circumference scales linearly with radius: $C = 2\pi r$.

18. What is the radius of $ $(x+6)^2$+$(y-1)^2$=49 $?

    $7$. Radius is $\sqrt{49} = 7$.

19. What is the definition of a circle with center $O$ and radius $r$ in the coordinate plane?

    The set of points $P$ with $OP = r$. A circle is the locus of points equidistant from its center.

20. What is the ratio of areas of similar circles with radii $r_1$ and $r_2$?

    $\frac{A_2}{A_1} = \left(\frac{r_2}{r_1}\right)^2$. Area scales with radius squared: $A = \pi r^2$.

21. What is the result of composing a rigid motion with a dilation regarding similarity?

    The composition is a similarity transformation. Similarity is preserved under composition with rigid motions.

22. What is the circle equation after translating $(x-1)^2 + (y+2)^2 = 9$ by $\langle 4, -3 \rangle$?

    $(x-5)^2 + (y+5)^2 = 9$. New center is $(1+4, -2-3) = (5,-5)$

23. Which option correctly proves two circles are similar using radii $r_1$ and $r_2$?

    Use dilation with $k = \frac{r_2}{r_1}$ (plus a rigid motion if needed). This ratio gives the proper scaling between the circles.

24. What is the equation of a circle centered at the origin with radius $r$?

    $x^2 + y^2 = r^2$. Special case when center is at origin $(0,0)$.

25. Find and correct the error: 'A dilation can send a circle to an ellipse.'

    Correct: A dilation sends a circle to a circle (or a point if $k=0$). Dilations preserve circular shape, never creating ellipses.

26. What property of a dilation describes how it changes all lengths?

    All lengths are multiplied by the scale factor $|k|$. Dilation uniformly scales all distances by the absolute value of $k$.

27. Identify the circumference ratio $\frac{C_2}{C_1}$ if radii satisfy $r_2 = 4r_1$.

    $\frac{C_2}{C_1} = 4$. Circumference ratio equals radius ratio.

28. What is the image of the center of a circle under a dilation with center $O$?

    The center maps to its dilated point from $O$. Center moves along the ray from dilation center $O$.

29. Identify the transformation mapping a circle with center $(0,0)$, radius $2$ to center $(5,-3)$, radius $6$.

    Dilate by $k = 3$, then translate by $\langle 5,-3 \rangle$. First scale radius $2$ to $6$, then move center.

30. What is the definition of a dilation with center $O$ and scale factor $k$?

    A map sending $P$ to $P'$ on ray $OP$ with $OP' = k\,OP$. Points move along rays from center $O$, scaled by factor $k$.

31. What is the key idea used to prove that all circles are similar?

    Any circle can be mapped to any other by a dilation and translation. Any circle can be transformed to match any other circle's size and position.

32. What is the scale factor that maps a circle of radius $r_1$ to radius $r_2$?

    $k = \frac{r_2}{r_1}$. Ratio of target radius to source radius gives the scaling needed.

33. What is the image radius if a circle of radius $\frac{3}{4}$ is dilated by $k = 8$?

    $6$. Multiply preimage radius by scale factor: $\frac{3}{4} \times 8$.

34. What statement about radii is true for any single circle?

    All radii of the same circle are congruent. All radii are line segments from center to circumference.

35. What is the scale factor mapping $(x-2)^2+(y+1)^2=9$ to a circle with radius $15$?

    $k = \frac{15}{3} = 5$. Target radius is $15$, source radius is $3$.

36. What is the image of a circle under a dilation in the plane?

    A circle (or a point if $k = 0$). Dilations map circles to circles (degenerate to point when $k=0$).

37. Which statement correctly relates arc lengths in similar circles with the same central angle $\theta$?

    $\frac{s_2}{s_1} = \frac{r_2}{r_1}$. Arc length scales linearly with radius for fixed angle.

38. Identify the area ratio $\frac{A_2}{A_1}$ if radii satisfy $r_2 = 3r_1$.

    $\frac{A_2}{A_1} = 9$. Area ratio equals radius ratio squared.

39. What happens to the radius $r$ of a circle under a dilation with factor $k$?

    The radius becomes $|k|r$. Dilation scales the radius by the absolute value of the factor.

40. Find the scale factor mapping a circle with diameter $18$ to a circle with diameter $30$.

    $k = \frac{30}{18} = \frac{5}{3}$. Scale factor equals ratio of diameters (or radii).

41. Which option is the correct scale factor mapping area $A_1$ to $A_2$ for similar circles?

    $k = \sqrt{\frac{A_2}{A_1}}$. Take square root since area scales quadratically.

42. What is the radius of $(x+6)^2+(y-1)^2=49$?

    $7$. Radius is $\sqrt{49} = 7$.

43. What is the scale factor if similar circles have circumferences $C_1=10\pi$ and $C_2=25\pi$?

    $k = \frac{25\pi}{10\pi} = \frac{5}{2}$. Circumference ratio gives the scale factor directly.

44. Identify the scale factor mapping $(x-1)^2+(y-2)^2=4$ to a circle with radius $14$.

    $k = \frac{14}{2} = 7$. Scale from radius $2$ up to radius $14$.

45. Identify the scale factor if two circles have areas $A_1=9\pi$ and $A_2=36\pi$.

    $k = 2$. Since $\frac{A_2}{A_1} = \frac{36\pi}{9\pi} = 4$, then $k = \sqrt{4} = 2$.

46. Which similarity transformation changes the size of a figure but preserves its shape?

    A dilation. Only dilation changes size; rigid motions preserve size.

47. Identify the scale factor that maps a circle of radius $4$ to a circle of radius $10$.

    $k = \frac{10}{4} = \frac{5}{2}$. Divide the target radius by the source radius.

48. What is the definition of similarity for two figures in the plane?

    Corresponding angles are equal and lengths are proportional. This defines when two shapes have the same angles but scaled proportionally.

49. What is the standard equation of a circle with center $(h,k)$ and radius $r$?

    $(x-h)^2 + (y-k)^2 = r^2$. Standard form derived from distance formula: $\sqrt{(x-h)^2+(y-k)^2}=r$.

50. What must be true about corresponding radii when one circle is the image of another under similarity?

    Their lengths are proportional by the same factor $k$. Similar figures have proportional corresponding linear dimensions.

51. What is the circle equation after dilating $(x-3)^2+(y-4)^2=25$ about its center by $k = 2$?

    $(x-3)^2 + (y-4)^2 = 100$. Dilation about center scales radius: $5 \times 2 = 10$.

52. What is the definition of the scale factor between two similar figures?

    The constant ratio of corresponding lengths, $k = \frac{\text{image}}{\text{preimage}}$. Scale factor compares corresponding lengths in image to preimage.

53. Which statement correctly relates chord lengths in similar circles for the same central angle $\theta$?

    $\frac{\text{chord}_2}{\text{chord}_1} = \frac{r_2}{r_1}$. Chord length scales linearly with radius for fixed central angle.

54. What is the image of a circle under a rotation about a point?

    A congruent circle. Rotation preserves all distances and angles.

55. Find and correct the error: 'All circles are congruent because they are similar.'

    Correct: All circles are similar; they are congruent only if radii are equal. Similarity doesn't require congruence; congruence needs equal radii.

56. Which option is the correct scale factor mapping circumference $C_1$ to $C_2$ for similar circles?

    $k = \frac{C_2}{C_1}$. Circumference scales linearly with the scale factor.

57. What is the image of a circle under a reflection across a line?

    A congruent circle. Reflection preserves all distances and angles.

58. What property of a dilation guarantees it preserves angle measures?

    A dilation is a similarity transformation, so angles are preserved. Similarity transformations preserve angle measures by definition.

59. What is the minimal similarity claim needed for CCSS.G-C.1 about circles?

    Any two circles are similar. This is the fundamental theorem: circle similarity is universal.

60. What is the circle equation after translating $(x-1)^2+(y+2)^2=9$ by $\langle 4,-3 \rangle$?

    $(x-5)^2 + (y+5)^2 = 9$. New center is $(1+4, -2-3) = (5,-5)$.

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