Question

you can transform circle v to circle v by translating it and then performing a dilation centered at v. find the translation rule and the scale factor of the dilation. simplify the scale factor and write it as a proper fraction, improper fraction, or whole number. translation: (x, y) → ( , ) scale factor:

Explanation:

Step1: Find the coordinates of centers

First, identify the center of circle \( V \) (let's call it \( (x_V, y_V) \)) and the center of circle \( V' \) ( \( (x_{V'}, y_{V'}) \) ). From the graph, center of \( V \) is at \( (1, 0) \) (wait, no, looking at the grid: the orange circle's center \( V \) is at \( (1, 0) \)? Wait, no, the yellow circle \( V' \) is at, let's see: the yellow circle is at \( (-7, -6) \)? Wait, no, let's check the grid again. Wait, the orange circle (larger) has center at \( (1, 0) \)? Wait, no, the grid: x-axis from -10 to 10, y-axis from -10 to 10. The yellow circle (smaller) is at, let's see, the center of \( V' \) (yellow) is at \( (-7, -6) \)? Wait, no, maybe I misread. Wait, the problem says "transform circle \( V \) to circle \( V' \)"? Wait, no, the problem says "transform circle \( V \) to circle \( V' \) by translating it and then performing a dilation centered at \( V' \)". Wait, no, the original circle is \( V \) (orange, larger) and \( V' \) is yellow (smaller). Wait, actually, to translate \( V \) to \( V' \), we need to find the vector from \( V \) to \( V' \). Let's find coordinates:

Center of \( V \) (orange circle): looking at the graph, the center is at \( (1, 0) \)? Wait, no, the orange circle is centered at \( (1, 0) \)? Wait, the yellow circle \( V' \) is at \( (-7, -6) \)? Wait, no, let's check the grid. Let's count the units. From \( V \) (orange center) to \( V' \) (yellow center): horizontal movement: from \( x = 1 \) to \( x = -7 \): that's \( -7 - 1 = -8 \) (so 8 units left). Vertical movement: from \( y = 0 \) to \( y = -6 \): that's \( -6 - 0 = -6 \) (6 units down). Wait, no, maybe I got the centers reversed. Wait, the problem says "transform circle \( V \) to circle \( V' \)", so we need to move \( V \) to \( V' \). So translation rule is \( (x, y) \to (x + h, y + k) \), where \( h = x_{V'} - x_V \), \( k = y_{V'} - y_V \).

Wait, let's re-express: Let's find the center of \( V \) (larger circle) and \( V' \) (smaller circle). From the graph:

- Center of \( V \) (orange) is at \( (1, 0) \) (since it's at x=1, y=0).
- Center of \( V' \) (yellow) is at \( (-7, -6) \).

So to translate \( V \) to \( V' \), the horizontal change \( h = -7 - 1 = -8 \), vertical change \( k = -6 - 0 = -6 \). Wait, but maybe I had it backwards. Wait, the problem says "transform circle \( V \) to circle \( V' \)", so we need to move \( V \) (original) to \( V' \) (target). So translation rule is \( (x, y) \to (x + h, y + k) \), where \( h = x_{V'} - x_V \), \( k = y_{V'} - y_V \). So \( x_V = 1 \), \( x_{V'} = -7 \), so \( h = -7 - 1 = -8 \). \( y_V = 0 \), \( y_{V'} = -6 \), so \( k = -6 - 0 = -6 \). Wait, but let's check again. Maybe the center of \( V \) is at \( (1, 0) \) and \( V' \) is at \( (-7, -6) \). So translation is \( (x, y) \to (x - 8, y - 6) \).

Now, for the scale factor: dilation centered at \( V' \), so we need the ratio of radii. Let's find the radius of \( V \) (larger circle) and \( V' \) (smaller circle). Radius of \( V \): from center \( (1, 0) \) to the edge, say, to \( x = 9 \) (since the circle goes to x=9, y=0: distance from \( (1,0) \) to \( (9,0) \) is \( 8 \) units, so radius \( r_V = 8 \). Radius of \( V' \): from center \( (-7, -6) \) to its edge, say, the circle is small, let's see, the diameter is, maybe 2 units? Wait, no, let's count the grid. Wait, the yellow circle ( \( V' \) ) has radius, let's see, from center \( (-7, -6) \) to the top, maybe 1 unit? Wait, no, maybe I made a mistake. Wait, the larger circle ( \( V \) ) has radius: from center \( (1, 0) \) to \( (9, 0) \): that's 8 units (since 9 - 1 = 8). The smaller circle ( \( V' \) ) has radius: from center \( (-7, -6) \) to, say, \( (-6, -6) \): that's 1 unit? Wait, no, maybe the radius of \( V' \) is 1, and \( V \) is 8? Wait, no, maybe the radius of \( V \) is 8 (from x=1 to x=9: 8 units), and radius of \( V' \) is 1 (from x=-7 to x=-6: 1 unit). Then scale factor (dilation from \( V \) to \( V' \), centered at \( V' \)): the scale factor \( s \) is \( \frac{\text{radius of } V'}{\text{radius of } V} \)? Wait, no: when dilating a figure centered at \( V' \), the scale factor is the ratio of the image ( \( V' \) ) radius to the original ( \( V \) ) radius. Wait, but the transformation is from \( V \) to \( V' \), so after translation, we dilate \( V \) (now at \( V' \) position) to \( V' \). Wait, no: first translate \( V \) to \( V' \), then dilate centered at \( V' \). So the original circle (before translation) is \( V \), after translation, it's at \( V' \) position (same size), then dilate to get \( V' \) (smaller). So the radius of the translated \( V \) (same as original \( V \)) is \( r_V \), and radius of \( V' \) is \( r_{V'} \). So scale factor \( s = \frac{r_{V'}}{r_V} \).

Wait, let's find radii:

- Radius of \( V \) (larger circle): distance from center \( (1, 0) \) to the rightmost point \( (9, 0) \): \( 9 - 1 = 8 \), so \( r_V = 8 \).
- Radius of \( V' \) (smaller circle): distance from center \( (-7, -6) \) to the rightmost point \( (-6, -6) \): \( -6 - (-7) = 1 \), so \( r_{V'} = 1 \).

Thus, scale factor \( s = \frac{r_{V'}}{r_V} = \frac{1}{8} \)? Wait, no, wait: when dilating the translated circle (which has radius 8) centered at \( V' \) to get \( V' \) (radius 1), the scale factor is \( \frac{1}{8} \). Wait, but maybe I mixed up the circles. Wait, the problem says "transform circle \( V \) to circle \( V' \)", so \( V \) is the original (larger), \( V' \) is the image (smaller). So translation: from \( V \) (center \( (1, 0) \)) to \( V' \) (center \( (-7, -6) \)): the translation rule is \( (x, y) \to (x + (-8), y + (-6)) \), so \( (x - 8, y - 6) \). Then scale factor: \( \frac{\text{radius of } V'}{\text{radius of } V} = \frac{1}{8} \)? Wait, no, wait, maybe the center of \( V \) is at \( (1, 0) \) and \( V' \) is at \( (-7, -6) \). Wait, let's recheck the coordinates:

Looking at the grid:

- Center of \( V \) (orange circle): x=1, y=0 (since it's at the intersection of x=1, y=0).
- Center of \( V' \) (yellow circle): x=-7, y=-6 (counting: from x=0, left 7 units: x=-7; from y=0, down 6 units: y=-6).

So translation: to move from \( (1, 0) \) to \( (-7, -6) \), we subtract 8 from x (1 - 8 = -7) and subtract 6 from y (0 - 6 = -6). So translation rule is \( (x, y) \to (x - 8, y - 6) \).

Now, radius of \( V \) (larger circle): from center \( (1, 0) \) to the top, y=8: distance is 8 units (from y=0 to y=8), so radius \( r_V = 8 \).

Radius of \( V' \) (smaller circle): from center \( (-7, -6) \) to the top, y=-5: distance is 1 unit (from y=-6 to y=-5), so radius \( r_{V'} = 1 \).

Thus, scale factor (dilation centered at \( V' \)) is \( \frac{r_{V'}}{r_V} = \frac{1}{8} \)? Wait, no, wait: when you dilate the translated circle (which has radius 8) centered at \( V' \) to get \( V' \) (radius 1), the scale factor is \( \frac{1}{8} \). Wait, but maybe I got the direction wrong. Wait, the problem says "transform circle \( V \) to circle \( V' \)", so \( V \) is the pre-image, \( V' \) is the image. So dilation scale factor is \( \frac{\text{image radius}}{\text{pre-image radius}} = \frac{1}{8} \).

Wait, but let's confirm:

- Translation: from \( V \) (center \( (1, 0) \)) to \( V' \) (center \( (-7, -6) \)): so \( h = -7 - 1 = -8 \), \( k = -6 - 0 = -6 \). So translation rule is \( (x, y) \to (x - 8, y - 6) \).

- Scale factor: radius of \( V \) is 8 (from center (1,0) to (9,0): 8 units), radius of \( V' \) is 1 (from center (-7,-6) to (-6,-6): 1 unit). So scale factor \( \frac{1}{8} \).

Wait, but maybe the center of \( V \) is at (1, 0) and \( V' \) is at (-7, -6). So:

Step2: Determine translation rule

Translation rule: \( (x, y) \to (x + (-8), y + (-6)) = (x - 8, y - 6) \).

Step3: Determine scale factor

Scale factor is \( \frac{\text{radius of } V'}{\text{radius of } V} = \frac{1}{8} \). Wait, but let's check the graph again. Wait, the larger circle (orange) has a radius that reaches from x=1 to x=9 (so 8 units), and the smaller circle (yellow) has a radius that reaches from x=-7 to x=-6 (1 unit). So yes, scale factor is \( \frac{1}{8} \).

Wait, but maybe I made a mistake in the center of \( V \). Wait, the orange circle is centered at (1, 0)? Let's check the y-axis: the orange circle goes up to y=8 and down to y=-8, so radius is 8, center at (1, 0). The yellow circle goes up to y=-5 and down to y=-7, so radius 1, center at (-7, -6). Yes, that makes sense.

So:

Translation: \( (x, y) \to (x - 8, y - 6) \)

Scale factor: \( \frac{1}{8} \)

Wait, but let's confirm the translation: from (1, 0) to (-7, -6): 1 - 8 = -7, 0 - 6 = -6. Correct.

Answer:

Translation: \( (x, y) \to (x - 8, y - 6) \) (so the blanks are -8 and -6)
Scale factor: \( \frac{1}{8} \)