Properties of Dilations and Scale Factor - Geometry
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What is the quickest construction sequence to inscribe a square using perpendicular diameters?
What is the quickest construction sequence to inscribe a square using perpendicular diameters?
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Draw a diameter, then a perpendicular diameter. Perpendicular diameters create four equally-spaced vertices for the square.
Draw a diameter, then a perpendicular diameter. Perpendicular diameters create four equally-spaced vertices for the square.
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Which chord length equals the radius in a circle, enabling a regular hexagon construction?
Which chord length equals the radius in a circle, enabling a regular hexagon construction?
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A chord of length $r$. A chord of length $r$ subtends a $60^\circ$ central angle in the circle.
A chord of length $r$. A chord of length $r$ subtends a $60^\circ$ central angle in the circle.
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What is the first step to construct an inscribed equilateral triangle if the center is known?
What is the first step to construct an inscribed equilateral triangle if the center is known?
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Construct a $120^\circ$ central partition. Equilateral triangle vertices are spaced $120^\circ$ apart around the center.
Construct a $120^\circ$ central partition. Equilateral triangle vertices are spaced $120^\circ$ apart around the center.
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Identify the polygon: an inscribed regular polygon with central angle $90^\circ$.
Identify the polygon: an inscribed regular polygon with central angle $90^\circ$.
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Square. Central angle $90^\circ$ corresponds to a 4-sided polygon.
Square. Central angle $90^\circ$ corresponds to a 4-sided polygon.
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Identify the polygon: an inscribed regular polygon with central angle $120^\circ$.
Identify the polygon: an inscribed regular polygon with central angle $120^\circ$.
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Equilateral triangle. Central angle $120^\circ$ corresponds to a 3-sided polygon.
Equilateral triangle. Central angle $120^\circ$ corresponds to a 3-sided polygon.
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What is the quickest construction sequence to inscribe an equilateral triangle using only a hexagon construction?
What is the quickest construction sequence to inscribe an equilateral triangle using only a hexagon construction?
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Construct hexagon, then connect every other vertex. Skip every other hexagon vertex to get $120^\circ$ triangle spacing.
Construct hexagon, then connect every other vertex. Skip every other hexagon vertex to get $120^\circ$ triangle spacing.
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Find and correct the error: "An equilateral triangle comes from connecting adjacent hexagon vertices." What is the correction?
Find and correct the error: "An equilateral triangle comes from connecting adjacent hexagon vertices." What is the correction?
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Connect every other hexagon vertex. Triangle vertices are every other hexagon vertex, not adjacent ones.
Connect every other hexagon vertex. Triangle vertices are every other hexagon vertex, not adjacent ones.
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Find and correct the error: "A square inscribed in a circle uses two parallel diameters." What is the correction?
Find and correct the error: "A square inscribed in a circle uses two parallel diameters." What is the correction?
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Use two perpendicular diameters. Square requires perpendicular diameters, not parallel ones.
Use two perpendicular diameters. Square requires perpendicular diameters, not parallel ones.
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Find and correct the error: "To inscribe a hexagon, set the compass to the diameter." What is the correction?
Find and correct the error: "To inscribe a hexagon, set the compass to the diameter." What is the correction?
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Set the compass to the radius, not the diameter. Hexagon construction requires compass set to radius, not diameter.
Set the compass to the radius, not the diameter. Hexagon construction requires compass set to radius, not diameter.
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What is the minimum number of distinct points on the circle needed to determine an inscribed regular hexagon?
What is the minimum number of distinct points on the circle needed to determine an inscribed regular hexagon?
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$6$. Six vertices are needed to define the corners of a hexagon.
$6$. Six vertices are needed to define the corners of a hexagon.
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What is the minimum number of distinct points on the circle needed to determine an inscribed equilateral triangle?
What is the minimum number of distinct points on the circle needed to determine an inscribed equilateral triangle?
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$3$. Three vertices are needed to define the corners of a triangle.
$3$. Three vertices are needed to define the corners of a triangle.
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What is the minimum number of distinct points on the circle needed to determine an inscribed square by connecting them?
What is the minimum number of distinct points on the circle needed to determine an inscribed square by connecting them?
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$4$. Four vertices are needed to define the corners of a square.
$4$. Four vertices are needed to define the corners of a square.
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Which option correctly describes how to mark the next vertex when inscribing a regular hexagon?
Which option correctly describes how to mark the next vertex when inscribing a regular hexagon?
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Draw an arc with radius $r$ from the current vertex. Arc of radius $r$ from current vertex locates the next hexagon vertex.
Draw an arc with radius $r$ from the current vertex. Arc of radius $r$ from current vertex locates the next hexagon vertex.
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What is the defining property of vertices of an inscribed equilateral triangle made from a hexagon?
What is the defining property of vertices of an inscribed equilateral triangle made from a hexagon?
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Vertices are $2$ hexagon steps apart on the circle. Skip one hexagon vertex to get triangle vertices $120^\circ$ apart.
Vertices are $2$ hexagon steps apart on the circle. Skip one hexagon vertex to get triangle vertices $120^\circ$ apart.
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What is the defining property of adjacent vertices of an inscribed regular hexagon relative to the radius?
What is the defining property of adjacent vertices of an inscribed regular hexagon relative to the radius?
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Adjacent vertices are one radius-chord apart. Each hexagon side is a chord of length equal to the circle 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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What is the defining property of the $4$ vertices of an inscribed square relative to the circle center?
What is the defining property of the $4$ vertices of an inscribed square relative to the circle center?
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They are endpoints of two perpendicular diameters. Square vertices lie where two perpendicular diameters meet the circle.
They are endpoints of two perpendicular diameters. Square vertices lie where two perpendicular diameters meet the circle.
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Which construction guarantees right angles needed for an inscribed square once one diameter is drawn?
Which construction guarantees right angles needed for an inscribed square once one diameter is drawn?
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Construct the perpendicular to the diameter at the center. Perpendicular to diameter creates the $90^\circ$ angles needed for square vertices.
Construct the perpendicular to the diameter at the center. Perpendicular to diameter creates the $90^\circ$ angles needed for square vertices.
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If the central angle between adjacent vertices is $120^\circ$, how many sides does the inscribed regular polygon have?
If the central angle between adjacent vertices is $120^\circ$, how many sides does the inscribed regular polygon have?
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$3$. Central angle $120^\circ$ means $\frac{360^\circ}{120^\circ} = 3$ sides.
$3$. Central angle $120^\circ$ means $\frac{360^\circ}{120^\circ} = 3$ sides.
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What points are connected to form an inscribed square after two perpendicular diameters are drawn?
What points are connected to form an inscribed square after two perpendicular diameters are drawn?
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Connect the $4$ diameter endpoints in order. The four endpoints of perpendicular diameters form a square when connected.
Connect the $4$ diameter endpoints in order. The four endpoints of perpendicular diameters form a square when connected.
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What is the central angle measure between adjacent vertices of a regular hexagon inscribed in a circle?
What is the central angle measure between adjacent vertices of a regular hexagon inscribed in a circle?
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$60^\circ$. Hexagon has 6 sides, so $\frac{360^\circ}{6} = 60^\circ$.
$60^\circ$. Hexagon has 6 sides, so $\frac{360^\circ}{6} = 60^\circ$.
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If a circle has radius $r$, what is the side length of an inscribed regular hexagon in terms of $r$?
If a circle has radius $r$, what is the side length of an inscribed regular hexagon in terms of $r$?
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$r$. Regular hexagon side length equals the radius of its circumscribed circle.
$r$. Regular hexagon side length equals the radius of its circumscribed circle.
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If the central angle between adjacent vertices is $90^\circ$, how many sides does the inscribed regular polygon have?
If the central angle between adjacent vertices is $90^\circ$, how many sides does the inscribed regular polygon have?
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$4$. Central angle $90^\circ$ means $\frac{360^\circ}{90^\circ} = 4$ sides.
$4$. Central angle $90^\circ$ means $\frac{360^\circ}{90^\circ} = 4$ sides.
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If the central angle between adjacent vertices is $60^\circ$, how many sides does the inscribed regular polygon have?
If the central angle between adjacent vertices is $60^\circ$, how many sides does the inscribed regular polygon have?
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$6$. Central angle $60^\circ$ means $\frac{360^\circ}{60^\circ} = 6$ sides.
$6$. Central angle $60^\circ$ means $\frac{360^\circ}{60^\circ} = 6$ sides.
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If you have an inscribed hexagon labeled $A,B,C,D,E,F$, which vertices form the other inscribed equilateral triangle?
If you have an inscribed hexagon labeled $A,B,C,D,E,F$, which vertices form the other inscribed equilateral triangle?
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$B,D,F$. Vertices $B, D, F$ are the alternate vertices, also $120^\circ$ apart.
$B,D,F$. Vertices $B, D, F$ are the alternate vertices, also $120^\circ$ apart.
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If you have an inscribed hexagon labeled $A,B,C,D,E,F$, which vertices form an inscribed equilateral triangle?
If you have an inscribed hexagon labeled $A,B,C,D,E,F$, which vertices form an inscribed equilateral triangle?
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$A,C,E$. Vertices $A, C, E$ are spaced $120^\circ$ apart, forming an equilateral triangle.
$A,C,E$. Vertices $A, C, E$ are spaced $120^\circ$ apart, forming an equilateral triangle.
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If you step off the radius as a chord and return to the start after $6$ steps, what polygon is formed?
If you step off the radius as a chord and return to the start after $6$ steps, what polygon is formed?
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Regular hexagon. Six radius-length steps around the circle create a regular hexagon.
Regular hexagon. Six radius-length steps around the circle create a regular hexagon.
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Which option is the correct central angle formula for an inscribed regular $n$-gon: $\frac{n}{360^\circ}$ or $\frac{360^\circ}{n}$?
Which option is the correct central angle formula for an inscribed regular $n$-gon: $\frac{n}{360^\circ}$ or $\frac{360^\circ}{n}$?
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$\frac{360^\circ}{n}$. Central angle formula divides full rotation by number of sides.
$\frac{360^\circ}{n}$. Central angle formula divides full rotation by number of sides.
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Which option correctly states how many equal arcs an equilateral triangle divides the circle into?
Which option correctly states how many equal arcs an equilateral triangle divides the circle into?
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$3$ equal arcs. Three vertices divide the circle into three equal arcs of $120^\circ$ each.
$3$ equal arcs. Three vertices divide the circle into three equal arcs of $120^\circ$ each.
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Which option correctly states how many equal arcs a square divides the circle into?
Which option correctly states how many equal arcs a square divides the circle into?
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$4$ equal arcs. Four vertices divide the circle into four equal arcs of $90^\circ$ each.
$4$ equal arcs. Four vertices divide the circle into four equal arcs of $90^\circ$ each.
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What is the central angle measure between adjacent vertices of a square inscribed in a circle?
What is the central angle measure between adjacent vertices of a square inscribed in a circle?
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$90^\circ$. Square has 4 sides, so $\frac{360^\circ}{4} = 90^\circ$.
$90^\circ$. Square has 4 sides, so $\frac{360^\circ}{4} = 90^\circ$.
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