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Sagot :
Let's find which of the given points lie on the perpendicular bisector of the segment between the points [tex]\((-8, 19)\)[/tex] and [tex]\((2, -7)\)[/tex].
1. Midpoint Calculation:
The midpoint of a segment between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \][/tex]
For the points [tex]\((-8, 19)\)[/tex] and [tex]\((2, -7)\)[/tex], the midpoint is:
[tex]\[ \left(\frac{-8 + 2}{2}, \frac{19 + (-7)}{2}\right) = \left(\frac{-6}{2}, \frac{12}{2}\right) = (-3, 6) \][/tex]
2. Slope of the Segment:
The slope [tex]\(m\)[/tex] of the segment between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((-8, 19)\)[/tex] and [tex]\((2, -7)\)[/tex], the slope is:
[tex]\[ m = \frac{-7 - 19}{2 - (-8)} = \frac{-26}{10} = -2.6 \][/tex]
3. Slope of the Perpendicular Bisector:
The slope of the perpendicular bisector is the negative reciprocal of the slope of the segment. If the slope of the segment is [tex]\(m\)[/tex], the slope of the perpendicular bisector is:
[tex]\[ m_\perp = -\frac{1}{m} \][/tex]
For the slope [tex]\(-2.6\)[/tex], the negative reciprocal is:
[tex]\[ m_\perp = -\frac{1}{-2.6} = \frac{1}{2.6} \approx 0.3846 \][/tex]
4. Equation of the Perpendicular Bisector:
The equation of a line in point-slope form is given by [tex]\(y - y_1 = m(x - x_1)\)[/tex]. For the perpendicular bisector through the midpoint [tex]\((-3, 6)\)[/tex] with slope [tex]\(0.3846\)[/tex], we have:
[tex]\[ y - 6 = 0.3846(x + 3) \][/tex]
5. Checking Each Point:
Substitute each point into the equation and see if the equation holds true:
- For the point [tex]\((-8, 19)\)[/tex]:
[tex]\[ 19 - 6 = 0.3846(-8 + 3) \implies 13 = 0.3846 \times (-5) \implies 13 \neq -1.923 \][/tex]
This point is not on the perpendicular bisector.
- For the point [tex]\((1, -8)\)[/tex]:
[tex]\[ -8 - 6 = 0.3846(1 + 3) \implies -14 = 0.3846 \times 4 \implies -14 \neq 1.5384 \][/tex]
This point is not on the perpendicular bisector.
- For the point [tex]\((0, 19)\)[/tex]:
[tex]\[ 19 - 6 = 0.3846(0 + 3) \implies 13 = 0.3846 \times 3 \implies 13 \neq 1.1538 \][/tex]
This point is not on the perpendicular bisector.
- For the point [tex]\((-5, 10)\)[/tex]:
[tex]\[ 10 - 6 = 0.3846(-5 + 3) \implies 4 = 0.3846 \times (-2) \implies 4 \neq -0.7692 \][/tex]
This point is not on the perpendicular bisector.
- For the point [tex]\((2, -7)\)[/tex]:
[tex]\[ -7 - 6 = 0.3846(2 + 3) \implies -13 = 0.3846 \times 5 \implies -13 \neq 1.923 \][/tex]
This point is not on the perpendicular bisector.
None of the given points [tex]\((-8, 19)\)[/tex], [tex]\((1, -8)\)[/tex], [tex]\((0, 19)\)[/tex], [tex]\((-5, 10)\)[/tex], and [tex]\((2, -7)\)[/tex] lie on the perpendicular bisector of the segment.
1. Midpoint Calculation:
The midpoint of a segment between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \][/tex]
For the points [tex]\((-8, 19)\)[/tex] and [tex]\((2, -7)\)[/tex], the midpoint is:
[tex]\[ \left(\frac{-8 + 2}{2}, \frac{19 + (-7)}{2}\right) = \left(\frac{-6}{2}, \frac{12}{2}\right) = (-3, 6) \][/tex]
2. Slope of the Segment:
The slope [tex]\(m\)[/tex] of the segment between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((-8, 19)\)[/tex] and [tex]\((2, -7)\)[/tex], the slope is:
[tex]\[ m = \frac{-7 - 19}{2 - (-8)} = \frac{-26}{10} = -2.6 \][/tex]
3. Slope of the Perpendicular Bisector:
The slope of the perpendicular bisector is the negative reciprocal of the slope of the segment. If the slope of the segment is [tex]\(m\)[/tex], the slope of the perpendicular bisector is:
[tex]\[ m_\perp = -\frac{1}{m} \][/tex]
For the slope [tex]\(-2.6\)[/tex], the negative reciprocal is:
[tex]\[ m_\perp = -\frac{1}{-2.6} = \frac{1}{2.6} \approx 0.3846 \][/tex]
4. Equation of the Perpendicular Bisector:
The equation of a line in point-slope form is given by [tex]\(y - y_1 = m(x - x_1)\)[/tex]. For the perpendicular bisector through the midpoint [tex]\((-3, 6)\)[/tex] with slope [tex]\(0.3846\)[/tex], we have:
[tex]\[ y - 6 = 0.3846(x + 3) \][/tex]
5. Checking Each Point:
Substitute each point into the equation and see if the equation holds true:
- For the point [tex]\((-8, 19)\)[/tex]:
[tex]\[ 19 - 6 = 0.3846(-8 + 3) \implies 13 = 0.3846 \times (-5) \implies 13 \neq -1.923 \][/tex]
This point is not on the perpendicular bisector.
- For the point [tex]\((1, -8)\)[/tex]:
[tex]\[ -8 - 6 = 0.3846(1 + 3) \implies -14 = 0.3846 \times 4 \implies -14 \neq 1.5384 \][/tex]
This point is not on the perpendicular bisector.
- For the point [tex]\((0, 19)\)[/tex]:
[tex]\[ 19 - 6 = 0.3846(0 + 3) \implies 13 = 0.3846 \times 3 \implies 13 \neq 1.1538 \][/tex]
This point is not on the perpendicular bisector.
- For the point [tex]\((-5, 10)\)[/tex]:
[tex]\[ 10 - 6 = 0.3846(-5 + 3) \implies 4 = 0.3846 \times (-2) \implies 4 \neq -0.7692 \][/tex]
This point is not on the perpendicular bisector.
- For the point [tex]\((2, -7)\)[/tex]:
[tex]\[ -7 - 6 = 0.3846(2 + 3) \implies -13 = 0.3846 \times 5 \implies -13 \neq 1.923 \][/tex]
This point is not on the perpendicular bisector.
None of the given points [tex]\((-8, 19)\)[/tex], [tex]\((1, -8)\)[/tex], [tex]\((0, 19)\)[/tex], [tex]\((-5, 10)\)[/tex], and [tex]\((2, -7)\)[/tex] lie on the perpendicular bisector of the segment.
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