Properties of Dilations and Scale Factor - Geometry

1. What is the quickest construction sequence to inscribe an equilateral triangle using only a hexagon construction?

   Construct hexagon, then connect every other vertex. Skip every other hexagon vertex to get $120^\circ$ triangle spacing.

2. If a circle has radius $r$, what is the side length of an inscribed regular hexagon in terms of $r$?

   $r$. Regular hexagon side length equals the radius of its circumscribed circle.

3. Which option correctly describes how to mark the next vertex when inscribing a regular hexagon?

   Draw an arc with radius $r$ from the current vertex. Arc of radius $r$ from current vertex locates the next hexagon vertex.

4. What is the first step to construct an inscribed square if the center of the circle is known?

   Draw a diameter through the center. One diameter provides two vertices that are $180^\circ$ apart.

5. What is the quickest construction sequence to inscribe a square using perpendicular diameters?

   Draw a diameter, then a perpendicular diameter. Perpendicular diameters create four equally-spaced vertices for the square.

6. What is the central-angle step size when using a regular hexagon to create an inscribed equilateral triangle?

   Skip one vertex, giving $120^\circ$ steps. Moving from one triangle vertex to the next skips $60^\circ$ to get $120^\circ$.

7. Find and correct the error: "To inscribe a hexagon, set the compass to the diameter." What is the correction?

   Set the compass to the radius, not the diameter. Hexagon construction requires compass set to radius, not diameter.

8. If a diameter has endpoints $A$ and $C$, and the perpendicular diameter has endpoints $B$ and $D$, what quadrilateral is $ABCD$?

   An inscribed square. Four endpoints of perpendicular diameters form a square.

9. Which option correctly states how many equal arcs a regular hexagon divides the circle into?

   $6$ equal arcs. Six vertices divide the circle into six equal arcs of $60^\circ$ each.

10. What points are connected to form an inscribed square after two perpendicular diameters are drawn?

    Connect the $4$ diameter endpoints in order. The four endpoints of perpendicular diameters form a square when connected.

11. What is the defining property of the $4$ vertices of an inscribed square relative to the circle center?

    They are endpoints of two perpendicular diameters. Square vertices lie where two perpendicular diameters meet the circle.

12. What is the minimum number of distinct points on the circle needed to determine an inscribed equilateral triangle?

    $3$. Three vertices are needed to define the corners of a triangle.

13. What is the central angle measure between adjacent vertices of an equilateral triangle inscribed in a circle?

    $120^\circ$. Triangle has 3 sides, so $\frac{360^\circ}{3} = 120^\circ$.

14. In an inscribed equilateral triangle, what is the angle measure of each interior angle of the triangle?

    $60^\circ$. All angles in an equilateral triangle measure $60^\circ$.

15. Find and correct the error: "An equilateral triangle comes from connecting adjacent hexagon vertices." What is the correction?

    Connect every other hexagon vertex. Triangle vertices are every other hexagon vertex, not adjacent ones.

16. What is the common name of the segment joining the circle center to a point on the circle?

    Radius. A radius connects the center to any point on the circle's edge.

17. What is the key step to obtain the other square vertices after drawing one diameter of the circle?

    Construct a perpendicular diameter. A perpendicular diameter creates four vertices spaced $90^\circ$ apart.

18. What is the relationship between a circle radius and a side of an inscribed regular hexagon?

    Side length $=$ radius. In a regular hexagon, the central angle is $60^\circ$, making each side equal to the radius.

19. Which regular polygon is obtained by selecting every other vertex of an inscribed regular hexagon?

    An inscribed equilateral triangle. Skipping every other hexagon vertex gives $120^\circ$ spacing for triangle vertices.

20. In an inscribed square, what is the angle measure of each interior angle of the square?

    $90^\circ$. All angles in a square are right angles measuring $90^\circ$.

21. If you have an inscribed hexagon labeled $A,B,C,D,E,F$, which vertices form an inscribed equilateral triangle?

    $A,C,E$. Vertices $A, C, E$ are spaced $120^\circ$ apart, forming an equilateral triangle.

22. What is the chord length you repeatedly copy to inscribe a regular hexagon in a circle of radius $r$?

    $r$. Hexagon sides equal the radius, so copy chord length $r$.

23. What is the first step to construct an inscribed equilateral triangle if the center is known?

    Construct a $120^\circ$ central partition. Equilateral triangle vertices are spaced $120^\circ$ apart around the center.

24. If the central angle between adjacent vertices is $60^\circ$, how many sides does the inscribed regular polygon have?

    $6$. Central angle $60^\circ$ means $\frac{360^\circ}{60^\circ} = 6$ sides.

25. Which inscribed regular polygon can be constructed by repeatedly marking off the radius as a chord?

    Regular hexagon. Only the hexagon has sides equal to the radius, making this construction work.

26. Which construction guarantees right angles needed for an inscribed square once one diameter is drawn?

    Construct the perpendicular to the diameter at the center. Perpendicular to diameter creates the $90^\circ$ angles needed for square vertices.

27. What value of $n$ gives a central angle of $120^\circ$ for an inscribed regular $n$-gon?

    $n=3$. Solving $\frac{360^\circ}{n} = 120^\circ$ gives $n = 3$.

28. What is the defining property of vertices of an inscribed equilateral triangle made from a hexagon?

    Vertices are $2$ hexagon steps apart on the circle. Skip one hexagon vertex to get triangle vertices $120^\circ$ apart.

29. If the central angle between adjacent vertices is $120^\circ$, how many sides does the inscribed regular polygon have?

    $3$. Central angle $120^\circ$ means $\frac{360^\circ}{120^\circ} = 3$ sides.

30. Which option correctly states how many equal arcs a square divides the circle into?

    $4$ equal arcs. Four vertices divide the circle into four equal arcs of $90^\circ$ each.

31. Identify the polygon: an inscribed regular polygon with central angle $60^\circ$.

    Regular hexagon. Central angle $60^\circ$ corresponds to a 6-sided polygon.

32. What value of $n$ gives a central angle of $60^\circ$ for an inscribed regular $n$-gon?

    $n=6$. Solving $\frac{360^\circ}{n} = 60^\circ$ gives $n = 6$.

33. Identify the polygon: an inscribed regular polygon with central angle $120^\circ$.

    Equilateral triangle. Central angle $120^\circ$ corresponds to a 3-sided polygon.

34. If the central angle between adjacent vertices is $90^\circ$, how many sides does the inscribed regular polygon have?

    $4$. Central angle $90^\circ$ means $\frac{360^\circ}{90^\circ} = 4$ sides.

35. In an inscribed regular hexagon, what is the angle measure of each interior angle of the hexagon?

    $120^\circ$. All interior angles in a regular hexagon measure $120^\circ$.

36. Identify the polygon: an inscribed regular polygon with central angle $90^\circ$.

    Square. Central angle $90^\circ$ corresponds to a 4-sided polygon.

37. What value of $n$ gives a central angle of $90^\circ$ for an inscribed regular $n$-gon?

    $n=4$. Solving $\frac{360^\circ}{n} = 90^\circ$ gives $n = 4$.

38. What is the formula for the central angle of a regular $n$-gon inscribed in a circle?

    $\frac{360^\circ}{n}$. Central angle equals $360^\circ$ divided by the number of sides.

39. What is the simplest method to locate all $6$ vertices of a regular hexagon inscribed in a circle?

    Step off $6$ radius chords on the circle. Each radius chord creates a $60^\circ$ arc, and $6 \times 60^\circ = 360^\circ$.

40. If you have an inscribed hexagon labeled $A,B,C,D,E,F$, which vertices form the other inscribed equilateral triangle?

    $B,D,F$. Vertices $B, D, F$ are the alternate vertices, also $120^\circ$ apart.

41. What is the central angle measure between adjacent vertices of a regular hexagon inscribed in a circle?

    $60^\circ$. Hexagon has 6 sides, so $\frac{360^\circ}{6} = 60^\circ$.

42. Which option correctly states how many equal arcs an equilateral triangle divides the circle into?

    $3$ equal arcs. Three vertices divide the circle into three equal arcs of $120^\circ$ each.

43. What is the minimum number of distinct points on the circle needed to determine an inscribed square by connecting them?

    $4$. Four vertices are needed to define the corners of a square.

44. Which option is the correct central angle formula for an inscribed regular $n$-gon: $\frac{n}{360^\circ}$ or $\frac{360^\circ}{n}$?

    $\frac{360^\circ}{n}$. Central angle formula divides full rotation by number of sides.

45. If you step off the radius as a chord and return to the start after $6$ steps, what polygon is formed?

    Regular hexagon. Six radius-length steps around the circle create a regular hexagon.

46. Which chord length equals the radius in a circle, enabling a regular hexagon construction?

    A chord of length $r$. A chord of length $r$ subtends a $60^\circ$ central angle in the circle.

47. What is the minimum number of distinct points on the circle needed to determine an inscribed regular hexagon?

    $6$. Six vertices are needed to define the corners of a hexagon.

48. What is the common name of a segment with endpoints on the circle?

    Chord. A chord is any line segment connecting two points on the circle.

49. What is the quickest construction sequence to inscribe a regular hexagon starting from one point on the circle?

    Set compass to radius and step off $6$ points. Radius-length arcs around the circle locate all six hexagon vertices.

50. Find and correct the error: "A square inscribed in a circle uses two parallel diameters." What is the correction?

    Use two perpendicular diameters. Square requires perpendicular diameters, not parallel ones.

51. What construction tool setting is used to step off vertices for an inscribed regular hexagon?

    Compass width $=$ circle radius. Setting compass to radius creates chords of length $r$, which equals hexagon side length.

52. What is the common name of a chord that passes through the center of the circle?

    Diameter. A diameter is the longest possible chord, passing through the center.

53. What is the central angle measure between adjacent vertices of a square inscribed in a circle?

    $90^\circ$. Square has 4 sides, so $\frac{360^\circ}{4} = 90^\circ$.

54. What is the defining property of adjacent vertices of an inscribed regular hexagon relative to the radius?

    Adjacent vertices are one radius-chord apart. Each hexagon side is a chord of length equal to the circle radius.

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