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To solve the problem of finding which points lie on the perpendicular bisector of the segment with endpoints [tex]\(A = (-8, 19)\)[/tex] and [tex]\(B = (1, -8)\)[/tex], we need to follow these steps:
1. Calculate the Midpoint of Segment AB:
The midpoint [tex]\(M\)[/tex] of the segment with endpoints [tex]\(A = (x_1, y_1)\)[/tex] and [tex]\(B = (x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Plugging in the coordinates:
[tex]\[ M = \left( \frac{-8 + 1}{2}, \frac{19 + (-8)}{2} \right) = \left( \frac{-7}{2}, \frac{11}{2} \right) = (-3.5, 5.5) \][/tex]
2. Find the Slope of AB:
The slope [tex]\( m_{AB} \)[/tex] of the line passing through points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is given by:
[tex]\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the coordinates:
[tex]\[ m_{AB} = \frac{-8 - 19}{1 - (-8)} = \frac{-27}{9} = -3 \][/tex]
3. Determine the Slope of the Perpendicular Bisector:
The slope of the perpendicular bisector is the negative reciprocal of the slope of AB:
[tex]\[ m_{\perpendicular} = -\frac{1}{m_{AB}} = -\frac{1}{-3} = \frac{1}{3} = 0.3333 \][/tex]
4. Find the Equation of the Perpendicular Bisector:
Using the midpoint [tex]\( M = (-3.5, 5.5) \)[/tex] and the slope [tex]\( m_{\perpendicular} = 0.3333 \)[/tex], we can determine the equation of the bisector in slope-intercept form [tex]\( y = mx + c \)[/tex]. Substitute [tex]\( M \)[/tex] into the slope-intercept formula [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
[tex]\[ 5.5 = 0.3333(-3.5) + c \implies 5.5 = -1.16665 + c \implies c = 5.5 + 1.16665 = 6.6667 \][/tex]
Thus, the equation of the perpendicular bisector is:
[tex]\[ y = 0.3333x + 6.6667 \][/tex]
5. Check Which Points Lie on the Perpendicular Bisector:
We now substitute each point into the equation [tex]\( y = 0.3333x + 6.6667 \)[/tex] to see if the equation holds:
- For [tex]\( (-8, 19) \)[/tex]:
[tex]\[ 19 \neq 0.3333(-8) + 6.6667 \implies 19 \neq -2.6664 + 6.6667 \][/tex]
[tex]\[ 19 \neq 4.0003 \quad (\text{False}) \][/tex]
- For [tex]\( (1, -8) \)[/tex]:
[tex]\[ -8 \neq 0.3333(1) + 6.6667 \implies -8 \neq 0.3333 + 6.6667 \][/tex]
[tex]\[ -8 \neq 7 \quad (\text{False}) \][/tex]
- For [tex]\( (0, 19) \)[/tex]:
[tex]\[ 19 \neq 0.3333(0) + 6.6667 \implies 19 \neq 6.6667 \][/tex]
[tex]\[ 19 \neq 6.6667 \quad (\text{False}) \][/tex]
- For [tex]\( (-5, 10) \)[/tex]:
[tex]\[ 10 \neq 0.3333(-5) + 6.6667 \implies 10 \neq -1.6665 + 6.6667 \][/tex]
[tex]\[ 10 \neq 5.0002 \quad (\text{False}) \][/tex]
- For [tex]\( (2, -7) \)[/tex]:
[tex]\[ -7 \neq 0.3333(2) + 6.6667 \implies -7 \neq 0.6666 + 6.6667 \][/tex]
[tex]\[ -7 \neq 7.3333 \quad (\text{False}) \][/tex]
Therefore, none of the given points [tex]\((-8, 19)\)[/tex], [tex]\((1, -8)\)[/tex], [tex]\((0, 19)\)[/tex], [tex]\((-5, 10)\)[/tex], [tex]\((2, -7)\)[/tex] lie on the perpendicular bisector of the segment with endpoints [tex]\( (-8, 19) \)[/tex] and [tex]\( (1, -8) \)[/tex].
1. Calculate the Midpoint of Segment AB:
The midpoint [tex]\(M\)[/tex] of the segment with endpoints [tex]\(A = (x_1, y_1)\)[/tex] and [tex]\(B = (x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Plugging in the coordinates:
[tex]\[ M = \left( \frac{-8 + 1}{2}, \frac{19 + (-8)}{2} \right) = \left( \frac{-7}{2}, \frac{11}{2} \right) = (-3.5, 5.5) \][/tex]
2. Find the Slope of AB:
The slope [tex]\( m_{AB} \)[/tex] of the line passing through points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is given by:
[tex]\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the coordinates:
[tex]\[ m_{AB} = \frac{-8 - 19}{1 - (-8)} = \frac{-27}{9} = -3 \][/tex]
3. Determine the Slope of the Perpendicular Bisector:
The slope of the perpendicular bisector is the negative reciprocal of the slope of AB:
[tex]\[ m_{\perpendicular} = -\frac{1}{m_{AB}} = -\frac{1}{-3} = \frac{1}{3} = 0.3333 \][/tex]
4. Find the Equation of the Perpendicular Bisector:
Using the midpoint [tex]\( M = (-3.5, 5.5) \)[/tex] and the slope [tex]\( m_{\perpendicular} = 0.3333 \)[/tex], we can determine the equation of the bisector in slope-intercept form [tex]\( y = mx + c \)[/tex]. Substitute [tex]\( M \)[/tex] into the slope-intercept formula [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
[tex]\[ 5.5 = 0.3333(-3.5) + c \implies 5.5 = -1.16665 + c \implies c = 5.5 + 1.16665 = 6.6667 \][/tex]
Thus, the equation of the perpendicular bisector is:
[tex]\[ y = 0.3333x + 6.6667 \][/tex]
5. Check Which Points Lie on the Perpendicular Bisector:
We now substitute each point into the equation [tex]\( y = 0.3333x + 6.6667 \)[/tex] to see if the equation holds:
- For [tex]\( (-8, 19) \)[/tex]:
[tex]\[ 19 \neq 0.3333(-8) + 6.6667 \implies 19 \neq -2.6664 + 6.6667 \][/tex]
[tex]\[ 19 \neq 4.0003 \quad (\text{False}) \][/tex]
- For [tex]\( (1, -8) \)[/tex]:
[tex]\[ -8 \neq 0.3333(1) + 6.6667 \implies -8 \neq 0.3333 + 6.6667 \][/tex]
[tex]\[ -8 \neq 7 \quad (\text{False}) \][/tex]
- For [tex]\( (0, 19) \)[/tex]:
[tex]\[ 19 \neq 0.3333(0) + 6.6667 \implies 19 \neq 6.6667 \][/tex]
[tex]\[ 19 \neq 6.6667 \quad (\text{False}) \][/tex]
- For [tex]\( (-5, 10) \)[/tex]:
[tex]\[ 10 \neq 0.3333(-5) + 6.6667 \implies 10 \neq -1.6665 + 6.6667 \][/tex]
[tex]\[ 10 \neq 5.0002 \quad (\text{False}) \][/tex]
- For [tex]\( (2, -7) \)[/tex]:
[tex]\[ -7 \neq 0.3333(2) + 6.6667 \implies -7 \neq 0.6666 + 6.6667 \][/tex]
[tex]\[ -7 \neq 7.3333 \quad (\text{False}) \][/tex]
Therefore, none of the given points [tex]\((-8, 19)\)[/tex], [tex]\((1, -8)\)[/tex], [tex]\((0, 19)\)[/tex], [tex]\((-5, 10)\)[/tex], [tex]\((2, -7)\)[/tex] lie on the perpendicular bisector of the segment with endpoints [tex]\( (-8, 19) \)[/tex] and [tex]\( (1, -8) \)[/tex].
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