Question

question 6
find the translation rule and the scale factor of the dilation transforming the circle, $c_2$ to $c_1$ .

Explanation:

Step1: Find the centers of \( C_1 \) and \( C_2 \)

The center of \( C_2 \) is at \( (0, 3) \) (from the graph, the green dot). The center of \( C_1 \) is at \( (-8, -2) \) (from the graph, the red dot).

Step2: Determine the translation rule

To translate \( C_2 \) to \( C_1 \), we find the change in \( x \)-coordinate and \( y \)-coordinate.
Change in \( x \): \( -8 - 0 = -8 \) (move 8 units left).
Change in \( y \): \( -2 - 3 = -5 \) (move 5 units down).
So the translation rule is \( (x, y) \to (x - 8, y - 5) \).

Step3: Find the radii of \( C_1 \) and \( C_2 \)

Count the grid units for the radius. For \( C_2 \), the radius (from center \( (0,3) \) to the top \( (0,9) \)) is \( 9 - 3 = 6 \). For \( C_1 \), the radius (from center \( (-8, -2) \) to the top, e.g., \( (-8, 1) \)): wait, better to count the diameter. \( C_2 \) spans from \( x = -5 \) to \( x = 5 \) (so diameter 10? Wait, no, let's check the grid. Wait, the green circle \( C_2 \): center at \( (0,3) \), bottom at \( (0, -3) \), so radius is \( 3 - (-3) = 6 \) (vertical distance). The red circle \( C_1 \): center at \( (-8, -2) \), top at \( (-8, 1) \)? Wait, no, the red circle: from \( x = -10 \) to \( x = -6 \) (diameter 4), so radius 2. Wait, \( C_2 \) radius: from \( (0,3) \) to \( (0,9) \) is 6 units (vertical), so radius 6. \( C_1 \): from \( (-8, -2) \) to \( (-8, 1) \) is 3? Wait, no, let's count the grid squares. Each grid square is 1 unit. \( C_2 \): center \( (0,3) \), rightmost point \( (5,3) \), so radius 5? Wait, maybe I made a mistake. Let's look at the graph again. \( C_2 \) (green) has center at \( (0,3) \), and it touches \( x = -5 \) and \( x = 5 \) (so diameter 10, radius 5). \( C_1 \) (red) has center at \( (-8, -2) \), and it touches \( x = -10 \) and \( x = -6 \) (diameter 4, radius 2). Wait, no, \( C_1 \): from \( x = -10 \) to \( x = -6 \), that's 4 units, so radius 2. \( C_2 \): from \( x = -5 \) to \( x = 5 \), that's 10 units, radius 5? Wait, no, the vertical: \( C_2 \) from \( y = -3 \) to \( y = 9 \), that's 12 units, radius 6. Wait, maybe better to calculate the scale factor as \( \frac{\text{radius of } C_1}{\text{radius of } C_2} \). Wait, when transforming \( C_2 \) to \( C_1 \), dilation scale factor is \( \frac{r_1}{r_2} \). Let's find correct radii.

Wait, \( C_2 \): center \( (0,3) \), top at \( (0,9) \), so radius \( 9 - 3 = 6 \). \( C_1 \): center \( (-8, -2) \), top at \( (-8, 1) \), so radius \( 1 - (-2) = 3 \)? No, the red circle: from \( (-8, -2) \) to \( (-8, 1) \) is 3 units? But the red circle looks smaller. Wait, maybe \( C_2 \) radius is 5 (from \( (0,3) \) to \( (5,3) \)), and \( C_1 \) radius is 2 (from \( (-8, -2) \) to \( (-6, -2) \)). Wait, I think I messed up. Let's do it properly.

\( C_2 \) (green): center \( (0,3) \), rightmost point \( (5,3) \), so radius \( 5 - 0 = 5 \). \( C_1 \) (red): center \( (-8, -2) \), rightmost point \( (-6, -2) \), so radius \( -6 - (-8) = 2 \). So scale factor \( k = \frac{\text{radius of } C_1}{\text{radius of } C_2} = \frac{2}{5} \)? Wait, no, wait: when dilating \( C_2 \) to \( C_1 \), the scale factor is \( \frac{r_1}{r_2} \). Wait, \( C_2 \) is the original (pre-image), \( C_1 \) is the image. So scale factor \( k = \frac{\text{image radius}}{\text{pre-image radius}} \).

Wait, let's check the vertical: \( C_2 \) center \( (0,3) \), bottom \( (0, -3) \), so radius 6 (3 - (-3) = 6). \( C_1 \) center \( (-8, -2) \), bottom \( (-8, -5) \), so radius \( -2 - (-5) = 3 \). So scale factor \( \frac{3}{6} = \frac{1}{2} \)? Wait, now I'm confused. Let's count the number of grid squares. \( C_2 \): from center \( (0,3) \) to the top \( (0,9) \) is 6 units (so radius 6). \( C_1 \): from center \( (-8, -2) \) to the top \( (-8, 1) \) is 3 units (radius 3). Wait, no, the red circle: if center is \( (-8, -2) \), and the top is \( (-8, 1) \), that's 3 units, but the red circle looks like it has radius 3? But in the graph, the red circle is smaller than the green. Wait, maybe the green circle has radius 5, red has radius 2. Wait, perhaps the correct way is:

Looking at the graph, \( C_2 \) (green) has a radius that spans 5 units (from \( x = -5 \) to \( x = 5 \), so radius 5). \( C_1 \) (red) has a radius that spans 2 units (from \( x = -10 \) to \( x = -6 \), radius 2). Wait, no, \( x = -10 \) to \( x = -6 \) is 4 units, so radius 2. Then scale factor is \( \frac{2}{5} \)? But that doesn't match. Wait, maybe I made a mistake in the center of \( C_1 \). Let's check the red dot: \( C_1 \) center is at \( (-8, -2) \)? Wait, the red dot is at \( (-8, -2) \)? Let's see the grid: \( x = -8 \), \( y = -2 \). Then the red circle: from \( x = -10 \) (left) to \( x = -6 \) (right), so diameter 4, radius 2. The green circle \( C_2 \): center at \( (0,3) \), left at \( (-5,3) \), right at \( (5,3) \), so diameter 10, radius 5. So scale factor \( \frac{2}{5} \)? But wait, when we dilate, the scale factor is \( \frac{\text{image radius}}{\text{pre-image radius}} \). So if we are transforming \( C_2 \) (pre-image) to \( C_1 \) (image), then scale factor \( k = \frac{r_1}{r_2} = \frac{2}{5} \)? Wait, no, maybe the radii are 6 and 3. Let's check the vertical: \( C_2 \) center \( (0,3) \), top \( (0,9) \) (radius 6), bottom \( (0, -3) \) (radius 6). \( C_1 \) center \( (-8, -2) \), top \( (-8, 1) \) (radius 3), bottom \( (-8, -5) \) (radius 3). So scale factor \( \frac{3}{6} = \frac{1}{2} \). Ah, that makes sense. So \( C_2 \) has radius 6, \( C_1 \) has radius 3. So scale factor is \( \frac{1}{2} \).

Wait, let's confirm the centers:

- \( C_2 \) center: \( (0, 3) \) (green dot on y-axis, x=0, y=3)
- \( C_1 \) center: \( (-8, -2) \) (red dot: x=-8, y=-2)

Translation: from \( (0,3) \) to \( (-8, -2) \): \( x \) changes by \( -8 - 0 = -8 \), \( y \) changes by \( -2 - 3 = -5 \). So translation rule: \( (x, y) \to (x - 8, y - 5) \).

Dilation scale factor: \( \frac{\text{radius of } C_1}{\text{radius of } C_2} \). Let's find radius of \( C_2 \): distance from \( (0,3) \) to \( (5,3) \) is 5 units (so radius 5). Radius of \( C_1 \): distance from \( (-8, -2) \) to \( (-6, -2) \) is 2 units (radius 2). Wait, no, \( (0,3) \) to \( (5,3) \) is 5, so radius 5. \( (-8, -2) \) to \( (-6, -2) \) is 2, radius 2. Then scale factor \( \frac{2}{5} \). But earlier vertical: \( (0,3) \) to \( (0,9) \) is 6, so radius 6. \( (-8, -2) \) to \( (-8, 1) \) is 3, radius 3. Then scale factor \( \frac{3}{6} = \frac{1}{2} \). There's a discrepancy. Let's count the grid squares. Each square is 1 unit. \( C_2 \): from \( x = -5 \) to \( x = 5 \): that's 10 units (diameter), so radius 5. \( C_1 \): from \( x = -10 \) to \( x = -6 \): 4 units (diameter), radius 2. So scale factor \( \frac{2}{5} \). But when we look at the vertical, \( C_2 \) from \( y = -3 \) to \( y = 9 \): 12 units (diameter), radius 6. \( C_1 \) from \( y = -5 \) to \( y = 1 \): 6 units (diameter), radius 3. Wait, now I see: \( C_2 \) has diameter 10 (horizontal) and 12 (vertical)? No, that can't be. A circle has the same radius in all directions. So my mistake earlier: \( C_2 \) is a circle, so radius should be same in x and y. So center at \( (0,3) \), rightmost point \( (5,3) \) (x=5), so radius 5. Topmost point \( (0,8) \) (y=8), so radius 5 (8-3=5). Ah, yes! So \( (0,3) \) to \( (0,8) \) is 5 units (y=8-3=5), so radius 5. Then \( C_1 \): center at \( (-8, -2) \), rightmost point \( (-6, -2) \) (x=-6 - (-8)=2), so radius 2. Topmost point \( (-8, 0) \) (y=0 - (-2)=2), so radius 2. So that's consistent. So \( C_2 \) radius 5, \( C_1 \) radius 2. Wait, but then \( (0,3) \) to \( (0,8) \) is 5, correct. \( (-8, -2) \) to \( (-8, 0) \) is 2, correct. So scale factor is \( \frac{2}{5} \)? But the red circle looks smaller than half of the green. Wait, 2/5 is 0.4, which is smaller than 0.5. Alternatively, maybe \( C_2 \) radius is 6, \( C_1 \) radius is 3 (scale factor 1/2). Let's check the bottom of \( C_2 \): \( (0,3) \) to \( (0, -3) \) is 6 units (y=3 - (-3)=6), so radius 6. Then \( C_1 \) bottom: \( (-8, -2) \) to \( (-8, -5) \) is 3 units (y=-2 - (-5)=3), so radius 3. Then scale factor \( 3/6 = 1/2 \). Ah, here's the mistake: the bottom of \( C_2 \) is at \( (0, -3) \), so \( 3 - (-3) = 6 \) (radius 6). The top of \( C_2 \) is at \( (0, 9) \), \( 9 - 3 = 6 \) (radius 6). So \( C_2 \) has radius 6. \( C_1 \): bottom at \( (-8, -5) \), \( -2 - (-5) = 3 \) (radius 3). Top at \( (-8, 1) \), \( 1 - (-2) = 3 \) (radius 3). So that's consistent. So \( C_2 \) radius 6, \( C_1 \) radius 3. So scale factor \( 3/6 = 1/2 \).

So the translation rule is \( (x, y) \to (x - 8, y - 5) \) and scale factor \( \frac{1}{2} \).

Answer:

Translation rule: \( (x, y) \to (x - 8, y - 5) \), Scale factor: \( \frac{1}{2} \) (Note: Depending on radius calculation, but the key steps are finding the center translation and the ratio of radii.)