Circle Similarity - Geometry
Card 1 of 30
Identify the scale factor mapping $x^2+y^2=25$ to $x^2+y^2=1$.
Identify the scale factor mapping $x^2+y^2=25$ to $x^2+y^2=1$.
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$k = \frac{1}{5}$. Scale from radius $5$ down to radius $1$.
$k = \frac{1}{5}$. Scale from radius $5$ down to radius $1$.
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What is a correct similarity mapping from the unit circle to a circle of radius $R$?
What is a correct similarity mapping from the unit circle to a circle of radius $R$?
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A dilation by $k = R$ (about the same center). Scale up by target radius from unit circle.
A dilation by $k = R$ (about the same center). Scale up by target radius from unit circle.
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What property of a dilation guarantees it preserves angle measures?
What property of a dilation guarantees it preserves angle measures?
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A dilation is a similarity transformation, so angles are preserved. Similarity transformations preserve angle measures by definition.
A dilation is a similarity transformation, so angles are preserved. Similarity transformations preserve angle measures by definition.
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What is the standard equation of a circle with center $(h,k)$ and radius $r$?
What is the standard equation of a circle with center $(h,k)$ and radius $r$?
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$(x-h)^2 + (y-k)^2 = r^2$. Standard form derived from distance formula: $\sqrt{(x-h)^2+(y-k)^2}=r$.
$(x-h)^2 + (y-k)^2 = r^2$. Standard form derived from distance formula: $\sqrt{(x-h)^2+(y-k)^2}=r$.
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What is the center of the circle $(x-8)^2+(y+3)^2=16$?
What is the center of the circle $(x-8)^2+(y+3)^2=16$?
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$(8,-3)$. Center coordinates are $(8, -3)$ from the standard form.
$(8,-3)$. Center coordinates are $(8, -3)$ from the standard form.
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What is the radius of $ (x+6)^2+(y-1)^2=49 $?
What is the radius of $ (x+6)^2+(y-1)^2=49 $?
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$7$. Radius is $\sqrt{49} = 7$.
$7$. Radius is $\sqrt{49} = 7$.
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Which option is the correct scale factor mapping area $A_1$ to $A_2$ for similar circles?
Which option is the correct scale factor mapping area $A_1$ to $A_2$ for similar circles?
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$k = \sqrt{\frac{A_2}{A_1}}$. Take square root since area scales quadratically.
$k = \sqrt{\frac{A_2}{A_1}}$. Take square root since area scales quadratically.
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Which option is the correct scale factor mapping circumference $C_1$ to $C_2$ for similar circles?
Which option is the correct scale factor mapping circumference $C_1$ to $C_2$ for similar circles?
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$k = \frac{C_2}{C_1}$. Circumference scales linearly with the scale factor.
$k = \frac{C_2}{C_1}$. Circumference scales linearly with the scale factor.
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Find the scale factor mapping a circle with diameter $18$ to a circle with diameter $30$.
Find the scale factor mapping a circle with diameter $18$ to a circle with diameter $30$.
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$k = \frac{30}{18} = \frac{5}{3}$. Scale factor equals ratio of diameters (or radii).
$k = \frac{30}{18} = \frac{5}{3}$. Scale factor equals ratio of diameters (or radii).
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What is the scale factor mapping $(x-2)^2+(y+1)^2=9$ to a circle with radius $15$?
What is the scale factor mapping $(x-2)^2+(y+1)^2=9$ to a circle with radius $15$?
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$k = \frac{15}{3} = 5$. Target radius is $15$, source radius is $3$.
$k = \frac{15}{3} = 5$. Target radius is $15$, source radius is $3$.
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What is the scale factor mapping radius $r_1$ to radius $r_2$ when $r_1=r_2$?
What is the scale factor mapping radius $r_1$ to radius $r_2$ when $r_1=r_2$?
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$k = 1$. Equal radii means no scaling needed.
$k = 1$. Equal radii means no scaling needed.
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What is the result of composing a rigid motion with a dilation regarding similarity?
What is the result of composing a rigid motion with a dilation regarding similarity?
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The composition is a similarity transformation. Similarity is preserved under composition with rigid motions.
The composition is a similarity transformation. Similarity is preserved under composition with rigid motions.
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Identify the transformation mapping a circle with center $(0,0)$, radius $2$ to center $(5,-3)$, radius $6$.
Identify the transformation mapping a circle with center $(0,0)$, radius $2$ to center $(5,-3)$, radius $6$.
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Dilate by $k = 3$, then translate by $\langle 5,-3 \rangle$. First scale radius $2$ to $6$, then move center.
Dilate by $k = 3$, then translate by $\langle 5,-3 \rangle$. First scale radius $2$ to $6$, then move center.
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What is the circle equation after dilating $(x-3)^2+(y-4)^2=25$ about its center by $k = 2$?
What is the circle equation after dilating $(x-3)^2+(y-4)^2=25$ about its center by $k = 2$?
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$(x-3)^2 + (y-4)^2 = 100$. Dilation about center scales radius: $5 \times 2 = 10$.
$(x-3)^2 + (y-4)^2 = 100$. Dilation about center scales radius: $5 \times 2 = 10$.
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What is the circle equation after dilating $x^2+y^2=16$ about the origin by $k = \frac{1}{2}$?
What is the circle equation after dilating $x^2+y^2=16$ about the origin by $k = \frac{1}{2}$?
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$x^2 + y^2 = 4$. Radius scales from $4$ to $2$: $(\frac{1}{2})^2 \times 16 = 4$.
$x^2 + y^2 = 4$. Radius scales from $4$ to $2$: $(\frac{1}{2})^2 \times 16 = 4$.
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What is the circle equation after translating $(x-1)^2+(y+2)^2=9$ by $\langle 4,-3 \rangle$?
What is the circle equation after translating $(x-1)^2+(y+2)^2=9$ by $\langle 4,-3 \rangle$?
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$(x-5)^2 + (y+5)^2 = 9$. New center is $(1+4, -2-3) = (5,-5)$.
$(x-5)^2 + (y+5)^2 = 9$. New center is $(1+4, -2-3) = (5,-5)$.
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What is the center after translating a circle centered at $(2,-1)$ by $\langle 5,4 \rangle$?
What is the center after translating a circle centered at $(2,-1)$ by $\langle 5,4 \rangle$?
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$(7,3)$. Add translation vector components: $(2+5, -1+4)$.
$(7,3)$. Add translation vector components: $(2+5, -1+4)$.
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What is the radius after dilating a circle of radius $5$ by $k = -2$?
What is the radius after dilating a circle of radius $5$ by $k = -2$?
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$10$. Take absolute value: $|{-2}| \times 5 = 10$.
$10$. Take absolute value: $|{-2}| \times 5 = 10$.
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What is the radius after dilating a circle of radius $7$ by $k = \frac{3}{2}$?
What is the radius after dilating a circle of radius $7$ by $k = \frac{3}{2}$?
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$\frac{21}{2}$. Multiply original radius by scale factor: $7 \times \frac{3}{2}$.
$\frac{21}{2}$. Multiply original radius by scale factor: $7 \times \frac{3}{2}$.
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What is the scale factor that maps radius $12$ to radius $3$?
What is the scale factor that maps radius $12$ to radius $3$?
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$k = \frac{3}{12} = \frac{1}{4}$. Scale factor is target divided by source: $\frac{3}{12}$.
$k = \frac{3}{12} = \frac{1}{4}$. Scale factor is target divided by source: $\frac{3}{12}$.
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Identify the scale factor that maps a circle of radius $4$ to a circle of radius $10$.
Identify the scale factor that maps a circle of radius $4$ to a circle of radius $10$.
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$k = \frac{10}{4} = \frac{5}{2}$. Divide the target radius by the source radius.
$k = \frac{10}{4} = \frac{5}{2}$. Divide the target radius by the source radius.
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Identify the similarity transformation that maps $x^2 + y^2 = 1$ to $x^2 + y^2 = 9$.
Identify the similarity transformation that maps $x^2 + y^2 = 1$ to $x^2 + y^2 = 9$.
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A dilation about the origin with $k = 3$. Radius increases from $1$ to $3$, so scale factor is $3$.
A dilation about the origin with $k = 3$. Radius increases from $1$ to $3$, so scale factor is $3$.
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Which similarity transformation changes the size of a figure but preserves its shape?
Which similarity transformation changes the size of a figure but preserves its shape?
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A dilation. Only dilation changes size; rigid motions preserve size.
A dilation. Only dilation changes size; rigid motions preserve size.
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What is the image of a circle under a reflection across a line?
What is the image of a circle under a reflection across a line?
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A congruent circle. Reflection preserves all distances and angles.
A congruent circle. Reflection preserves all distances and angles.
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What is the image of a circle under a rotation about a point?
What is the image of a circle under a rotation about a point?
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A congruent circle. Rotation preserves all distances and angles.
A congruent circle. Rotation preserves all distances and angles.
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What is the image of a circle under a translation by vector $\langle a,b \rangle$?
What is the image of a circle under a translation by vector $\langle a,b \rangle$?
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A congruent circle with its center shifted by $\langle a,b \rangle$. Translation preserves size and shape, only changes position.
A congruent circle with its center shifted by $\langle a,b \rangle$. Translation preserves size and shape, only changes position.
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What is the image of the center of a circle under a dilation with center $O$?
What is the image of the center of a circle under a dilation with center $O$?
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The center maps to its dilated point from $O$. Center moves along the ray from dilation center $O$.
The center maps to its dilated point from $O$. Center moves along the ray from dilation center $O$.
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What is the image of a circle under a dilation in the plane?
What is the image of a circle under a dilation in the plane?
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A circle (or a point if $k = 0$). Dilations map circles to circles (degenerate to point when $k=0$).
A circle (or a point if $k = 0$). Dilations map circles to circles (degenerate to point when $k=0$).
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What happens to the radius $r$ of a circle under a dilation with factor $k$?
What happens to the radius $r$ of a circle under a dilation with factor $k$?
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The radius becomes $|k|r$. Dilation scales the radius by the absolute value of the factor.
The radius becomes $|k|r$. Dilation scales the radius by the absolute value of the factor.
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What is the scale factor that maps a circle of radius $r_1$ to radius $r_2$?
What is the scale factor that maps a circle of radius $r_1$ to radius $r_2$?
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$k = \frac{r_2}{r_1}$. Ratio of target radius to source radius gives the scaling needed.
$k = \frac{r_2}{r_1}$. Ratio of target radius to source radius gives the scaling needed.
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