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To find the coordinates of the center of the circle that passes through the points [tex]\((-5,0)\)[/tex], [tex]\((1,-12)\)[/tex], and [tex]\((21,-2)\)[/tex], we need to follow a series of steps involving geometry and algebra. Here’s a detailed, step-by-step solution:
1. Identify the Midpoints of the Segments:
- For the segment joining [tex]\((-5,0)\)[/tex] and [tex]\((1,-12)\)[/tex]:
[tex]\[ \text{Midpoint} = \left( \frac{-5 + 1}{2}, \frac{0 + (-12)}{2} \right) = (-2, -6) \][/tex]
- For the segment joining [tex]\((1,-12)\)[/tex] and [tex]\((21,-2)\)[/tex]:
[tex]\[ \text{Midpoint} = \left( \frac{1 + 21}{2}, \frac{-12 + (-2)}{2} \right) = (11, -7) \][/tex]
2. Calculate the Slopes of these Segments:
- For the segment [tex]\((-5,0)\)[/tex] to [tex]\((1,-12)\)[/tex]:
[tex]\[ m_1 = \frac{-12 - 0}{1 - (-5)} = \frac{-12}{6} = -2 \][/tex]
- For the segment [tex]\((1,-12)\)[/tex] to [tex]\((21,-2)\)[/tex]:
[tex]\[ m_2 = \frac{-2 - (-12)}{21 - 1} = \frac{10}{20} = 0.5 \][/tex]
3. Determine the Slopes of the Perpendicular Bisectors:
- The slope of the perpendicular bisector of the first segment (perpendicular to [tex]\(m_1\)[/tex]) is:
[tex]\[ \text{Slope} = -\frac{1}{m_1} = -\frac{1}{-2} = 0.5 \][/tex]
- The slope of the perpendicular bisector of the second segment (perpendicular to [tex]\(m_2\)[/tex]) is:
[tex]\[ \text{Slope} = -\frac{1}{m_2} = -\frac{1}{0.5} = -2 \][/tex]
4. Write the Equations of the Perpendicular Bisectors:
- For the perpendicular bisector of the first segment, passing through [tex]\((-2, -6)\)[/tex]:
[tex]\[ y + 6 = 0.5 (x + 2) \][/tex]
Simplifying:
[tex]\[ y = 0.5x - 5 \][/tex]
- For the perpendicular bisector of the second segment, passing through [tex]\((11, -7)\)[/tex]:
[tex]\[ y + 7 = -2(x - 11) \][/tex]
Simplifying:
[tex]\[ y = -2x + 15 \][/tex]
5. Solve the System of Linear Equations:
- Equate the two equations of the perpendicular bisectors to find their intersection, which is the center of the circle:
[tex]\[ 0.5x - 5 = -2x + 15 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ 0.5x + 2x = 15 + 5 \][/tex]
[tex]\[ 2.5x = 20 \][/tex]
[tex]\[ x = 8 \][/tex]
- Substitute [tex]\(x = 8\)[/tex] into one of the original equations to find [tex]\(y\)[/tex]:
[tex]\[ y = 0.5(8) - 5 \][/tex]
[tex]\[ y = 4 - 5 \][/tex]
[tex]\[ y = -1 \][/tex]
Thus, the coordinates of the center of the circle passing through the three given points are [tex]\((8, -1)\)[/tex].
1. Identify the Midpoints of the Segments:
- For the segment joining [tex]\((-5,0)\)[/tex] and [tex]\((1,-12)\)[/tex]:
[tex]\[ \text{Midpoint} = \left( \frac{-5 + 1}{2}, \frac{0 + (-12)}{2} \right) = (-2, -6) \][/tex]
- For the segment joining [tex]\((1,-12)\)[/tex] and [tex]\((21,-2)\)[/tex]:
[tex]\[ \text{Midpoint} = \left( \frac{1 + 21}{2}, \frac{-12 + (-2)}{2} \right) = (11, -7) \][/tex]
2. Calculate the Slopes of these Segments:
- For the segment [tex]\((-5,0)\)[/tex] to [tex]\((1,-12)\)[/tex]:
[tex]\[ m_1 = \frac{-12 - 0}{1 - (-5)} = \frac{-12}{6} = -2 \][/tex]
- For the segment [tex]\((1,-12)\)[/tex] to [tex]\((21,-2)\)[/tex]:
[tex]\[ m_2 = \frac{-2 - (-12)}{21 - 1} = \frac{10}{20} = 0.5 \][/tex]
3. Determine the Slopes of the Perpendicular Bisectors:
- The slope of the perpendicular bisector of the first segment (perpendicular to [tex]\(m_1\)[/tex]) is:
[tex]\[ \text{Slope} = -\frac{1}{m_1} = -\frac{1}{-2} = 0.5 \][/tex]
- The slope of the perpendicular bisector of the second segment (perpendicular to [tex]\(m_2\)[/tex]) is:
[tex]\[ \text{Slope} = -\frac{1}{m_2} = -\frac{1}{0.5} = -2 \][/tex]
4. Write the Equations of the Perpendicular Bisectors:
- For the perpendicular bisector of the first segment, passing through [tex]\((-2, -6)\)[/tex]:
[tex]\[ y + 6 = 0.5 (x + 2) \][/tex]
Simplifying:
[tex]\[ y = 0.5x - 5 \][/tex]
- For the perpendicular bisector of the second segment, passing through [tex]\((11, -7)\)[/tex]:
[tex]\[ y + 7 = -2(x - 11) \][/tex]
Simplifying:
[tex]\[ y = -2x + 15 \][/tex]
5. Solve the System of Linear Equations:
- Equate the two equations of the perpendicular bisectors to find their intersection, which is the center of the circle:
[tex]\[ 0.5x - 5 = -2x + 15 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ 0.5x + 2x = 15 + 5 \][/tex]
[tex]\[ 2.5x = 20 \][/tex]
[tex]\[ x = 8 \][/tex]
- Substitute [tex]\(x = 8\)[/tex] into one of the original equations to find [tex]\(y\)[/tex]:
[tex]\[ y = 0.5(8) - 5 \][/tex]
[tex]\[ y = 4 - 5 \][/tex]
[tex]\[ y = -1 \][/tex]
Thus, the coordinates of the center of the circle passing through the three given points are [tex]\((8, -1)\)[/tex].
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