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To determine which circles lie completely within the third quadrant, we need to evaluate each circle based on:
1. The center of the circle.
2. The radius of the circle.
3. The position of the circle relative to the third quadrant.
The third quadrant is defined by [tex]\(x < 0\)[/tex] and [tex]\(y < 0\)[/tex].
Let's analyze each circle one by one.
### Circle A: [tex]\((x+5)^2 + (y+0)^2 = 7\)[/tex]
- Center: [tex]\((-5, 0)\)[/tex]
- Radius: [tex]\(\sqrt{7}\)[/tex]
The center [tex]\((-5, 0)\)[/tex] is on the negative x-axis but not in the third quadrant because [tex]\(y = 0\)[/tex]. Therefore, this circle does not lie completely within the third quadrant.
### Circle B: [tex]\((x+12)^2 + (y+9)^2 = 9\)[/tex]
- Center: [tex]\((-12, -9)\)[/tex]
- Radius: [tex]\(\sqrt{9} = 3\)[/tex]
The center [tex]\((-12, -9)\)[/tex] lies within the third quadrant since [tex]\(-12 < 0\)[/tex] and [tex]\(-9 < 0\)[/tex]. To check if the circle lies completely within the third quadrant, we need to ensure that the entire circle remains within this region.
- The circle's boundary extends 3 units in all directions from the center.
- In the x-direction, it ranges from [tex]\(-12 - 3 = -15\)[/tex] to [tex]\(-12 + 3 = -9\)[/tex].
- In the y-direction, it ranges from [tex]\(-9 - 3 = -12\)[/tex] to [tex]\(-9 + 3 = -6\)[/tex].
Since all these x and y values are negative, the circle lies completely within the third quadrant.
### Circle C: [tex]\((x+3)^2 + (y+9)^2 = 82\)[/tex]
- Center: [tex]\((-3, -9)\)[/tex]
- Radius: [tex]\(\sqrt{82}\)[/tex]
The center [tex]\((-3, -9)\)[/tex] lies within the third quadrant since [tex]\(-3 < 0\)[/tex] and [tex]\(-9 < 0\)[/tex]. Let's check the boundaries:
- The circle's boundary extends [tex]\(\sqrt{82}\)[/tex] units in all directions from the center.
- [tex]\(\sqrt{82} \approx 9.06\)[/tex]
- In the x-direction, it ranges from [tex]\(-3 - 9.06 \approx -12.06\)[/tex] to [tex]\(-3 + 9.06 \approx 6.06\)[/tex].
- In the y-direction, it ranges from [tex]\(-9 - 9.06 \approx -18.06\)[/tex] to [tex]\(-9 + 9.06 \approx 0.06\)[/tex].
Since the circle extends into the positive x-direction, it does not lie completely within the third quadrant.
### Circle D: [tex]\((x+7)^2 + (y+7)^2 = 4\)[/tex]
- Center: [tex]\((-7, -7)\)[/tex]
- Radius: [tex]\(\sqrt{4} = 2\)[/tex]
The center [tex]\((-7, -7)\)[/tex] lies within the third quadrant since [tex]\(-7 < 0\)[/tex] and [tex]\(-7 < 0\)[/tex]. Let's check the boundaries:
- The circle's boundary extends 2 units in all directions from the center.
- In the x-direction, it ranges from [tex]\(-7 - 2 = -9\)[/tex] to [tex]\(-7 + 2 = -5\)[/tex].
- In the y-direction, it ranges from [tex]\(-7 - 2 = -9\)[/tex] to [tex]\(-7 + 2 = -5\)[/tex].
Since all these x and y values are negative, the circle lies completely within the third quadrant.
### Conclusion
The circles that lie completely within the third quadrant are:
- Circle B: [tex]\((x+12)^2 + (y+9)^2 = 9\)[/tex]
- Circle D: [tex]\((x+7)^2 + (y+7)^2 = 4\)[/tex]
Thus, the answer is:
[tex]\[ B \text{ and } D \][/tex]
1. The center of the circle.
2. The radius of the circle.
3. The position of the circle relative to the third quadrant.
The third quadrant is defined by [tex]\(x < 0\)[/tex] and [tex]\(y < 0\)[/tex].
Let's analyze each circle one by one.
### Circle A: [tex]\((x+5)^2 + (y+0)^2 = 7\)[/tex]
- Center: [tex]\((-5, 0)\)[/tex]
- Radius: [tex]\(\sqrt{7}\)[/tex]
The center [tex]\((-5, 0)\)[/tex] is on the negative x-axis but not in the third quadrant because [tex]\(y = 0\)[/tex]. Therefore, this circle does not lie completely within the third quadrant.
### Circle B: [tex]\((x+12)^2 + (y+9)^2 = 9\)[/tex]
- Center: [tex]\((-12, -9)\)[/tex]
- Radius: [tex]\(\sqrt{9} = 3\)[/tex]
The center [tex]\((-12, -9)\)[/tex] lies within the third quadrant since [tex]\(-12 < 0\)[/tex] and [tex]\(-9 < 0\)[/tex]. To check if the circle lies completely within the third quadrant, we need to ensure that the entire circle remains within this region.
- The circle's boundary extends 3 units in all directions from the center.
- In the x-direction, it ranges from [tex]\(-12 - 3 = -15\)[/tex] to [tex]\(-12 + 3 = -9\)[/tex].
- In the y-direction, it ranges from [tex]\(-9 - 3 = -12\)[/tex] to [tex]\(-9 + 3 = -6\)[/tex].
Since all these x and y values are negative, the circle lies completely within the third quadrant.
### Circle C: [tex]\((x+3)^2 + (y+9)^2 = 82\)[/tex]
- Center: [tex]\((-3, -9)\)[/tex]
- Radius: [tex]\(\sqrt{82}\)[/tex]
The center [tex]\((-3, -9)\)[/tex] lies within the third quadrant since [tex]\(-3 < 0\)[/tex] and [tex]\(-9 < 0\)[/tex]. Let's check the boundaries:
- The circle's boundary extends [tex]\(\sqrt{82}\)[/tex] units in all directions from the center.
- [tex]\(\sqrt{82} \approx 9.06\)[/tex]
- In the x-direction, it ranges from [tex]\(-3 - 9.06 \approx -12.06\)[/tex] to [tex]\(-3 + 9.06 \approx 6.06\)[/tex].
- In the y-direction, it ranges from [tex]\(-9 - 9.06 \approx -18.06\)[/tex] to [tex]\(-9 + 9.06 \approx 0.06\)[/tex].
Since the circle extends into the positive x-direction, it does not lie completely within the third quadrant.
### Circle D: [tex]\((x+7)^2 + (y+7)^2 = 4\)[/tex]
- Center: [tex]\((-7, -7)\)[/tex]
- Radius: [tex]\(\sqrt{4} = 2\)[/tex]
The center [tex]\((-7, -7)\)[/tex] lies within the third quadrant since [tex]\(-7 < 0\)[/tex] and [tex]\(-7 < 0\)[/tex]. Let's check the boundaries:
- The circle's boundary extends 2 units in all directions from the center.
- In the x-direction, it ranges from [tex]\(-7 - 2 = -9\)[/tex] to [tex]\(-7 + 2 = -5\)[/tex].
- In the y-direction, it ranges from [tex]\(-7 - 2 = -9\)[/tex] to [tex]\(-7 + 2 = -5\)[/tex].
Since all these x and y values are negative, the circle lies completely within the third quadrant.
### Conclusion
The circles that lie completely within the third quadrant are:
- Circle B: [tex]\((x+12)^2 + (y+9)^2 = 9\)[/tex]
- Circle D: [tex]\((x+7)^2 + (y+7)^2 = 4\)[/tex]
Thus, the answer is:
[tex]\[ B \text{ and } D \][/tex]
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