Answer :
To determine which points lie on the perpendicular bisector of the given segment, we need to go through several steps: finding the midpoint of the segment, calculating the slope of the segment, determining the slope of the perpendicular bisector, and finally checking each point to see whether it satisfies the equation of the perpendicular bisector.
1. Finding the Midpoint:
The endpoints of the segment are [tex]\((-8, 19)\)[/tex] and [tex]\((2, -7)\)[/tex]. The midpoint [tex]\(M\)[/tex] of the segment is calculated as follows:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Substituting the given points:
[tex]\[ M = \left( \frac{-8 + 2}{2}, \frac{19 - 7}{2} \right) = \left( \frac{-6}{2}, \frac{12}{2} \right) = (-3, 6) \][/tex]
2. Calculating the Slope of the Segment:
The slope [tex]\(m\)[/tex] of the segment connecting [tex]\((-8, 19)\)[/tex] and [tex]\((2, -7)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values:
[tex]\[ m = \frac{-7 - 19}{2 + 8} = \frac{-26}{10} = -2.6 \][/tex]
3. Determining the Slope of the Perpendicular Bisector:
The slope of the perpendicular bisector is the negative reciprocal of the slope of the segment. Therefore, if the slope [tex]\(m\)[/tex] of the segment is [tex]\(-2.6\)[/tex], then the slope [tex]\(m_{\perp}\)[/tex] of the perpendicular bisector is:
[tex]\[ m_{\perp} = -\frac{1}{-2.6} = 0.3846153846153846 \][/tex]
4. Equation of the Perpendicular Bisector:
The equation of the perpendicular bisector can be written using point-slope form [tex]\(y - y_1 = m(x - x_1)\)[/tex]:
[tex]\[ y - 6 = 0.3846153846153846(x + 3) \][/tex]
Simplifying,
[tex]\[ y - 6 = 0.3846153846153846x + 1.1538461538461537 \][/tex]
[tex]\[ y = 0.3846153846153846x + 7.153846153846154 \][/tex]
5. Checking Each Given Point:
We need to check if each of the given points satisfies the equation [tex]\(y = 0.3846153846153846x + 7.153846153846154\)[/tex].
a. Point [tex]\((-8, 19)\)[/tex]
[tex]\[ y = 0.3846153846153846(-8) + 7.153846153846154 = -3.076923076923077 + 7.153846153846154 = 4.077 \][/tex]
19 ≠ 4.077 → The point [tex]\((-8, 19)\)[/tex] is not on the perpendicular bisector.
b. Point [tex]\((1, -8)\)[/tex]
[tex]\[ y = 0.3846153846153846(1) + 7.153846153846154 = 0.3846153846153846 + 7.153846153846154 = 7.538 \][/tex]
[tex]\(-8 ≠ 7.538 \rightarrow The point (1, -8) is not on the perpendicular bisector. c. Point \((0, 19)\)[/tex]
[tex]\[ y = 0.3846153846153846(0) + 7.153846153846154 = 7.153846153846154 \][/tex]
19 ≠ 7.153846153846154 → The point [tex]\((0, 19)\)[/tex] is not on the perpendicular bisector.
d. Point [tex]\((-5, 10)\)[/tex]
[tex]\[ y = 0.3846153846153846(-5) + 7.153846153846154 = -1.923076923076923 + 7.153846153846154 = 5.231 \][/tex]
10 ≠ 5.231 → The point [tex]\((-5, 10)\)[/tex] is not on the perpendicular bisector.
e. Point [tex]\((2, -7)\)[/tex]
[tex]\[ y = 0.3846153846153846(2) + 7.153846153846154 = 0.7692307692307692 + 7.153846153846154 = 7.923 \][/tex]
-7 ≠ 7.923 → The point [tex]\((2, -7)\)[/tex] is not on the perpendicular bisector.
Based on these calculations, none of the given points lie on the perpendicular bisector of the segment connecting [tex]\((-8, 19)\)[/tex] and [tex]\((2, -7)\)[/tex].
1. Finding the Midpoint:
The endpoints of the segment are [tex]\((-8, 19)\)[/tex] and [tex]\((2, -7)\)[/tex]. The midpoint [tex]\(M\)[/tex] of the segment is calculated as follows:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Substituting the given points:
[tex]\[ M = \left( \frac{-8 + 2}{2}, \frac{19 - 7}{2} \right) = \left( \frac{-6}{2}, \frac{12}{2} \right) = (-3, 6) \][/tex]
2. Calculating the Slope of the Segment:
The slope [tex]\(m\)[/tex] of the segment connecting [tex]\((-8, 19)\)[/tex] and [tex]\((2, -7)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values:
[tex]\[ m = \frac{-7 - 19}{2 + 8} = \frac{-26}{10} = -2.6 \][/tex]
3. Determining the Slope of the Perpendicular Bisector:
The slope of the perpendicular bisector is the negative reciprocal of the slope of the segment. Therefore, if the slope [tex]\(m\)[/tex] of the segment is [tex]\(-2.6\)[/tex], then the slope [tex]\(m_{\perp}\)[/tex] of the perpendicular bisector is:
[tex]\[ m_{\perp} = -\frac{1}{-2.6} = 0.3846153846153846 \][/tex]
4. Equation of the Perpendicular Bisector:
The equation of the perpendicular bisector can be written using point-slope form [tex]\(y - y_1 = m(x - x_1)\)[/tex]:
[tex]\[ y - 6 = 0.3846153846153846(x + 3) \][/tex]
Simplifying,
[tex]\[ y - 6 = 0.3846153846153846x + 1.1538461538461537 \][/tex]
[tex]\[ y = 0.3846153846153846x + 7.153846153846154 \][/tex]
5. Checking Each Given Point:
We need to check if each of the given points satisfies the equation [tex]\(y = 0.3846153846153846x + 7.153846153846154\)[/tex].
a. Point [tex]\((-8, 19)\)[/tex]
[tex]\[ y = 0.3846153846153846(-8) + 7.153846153846154 = -3.076923076923077 + 7.153846153846154 = 4.077 \][/tex]
19 ≠ 4.077 → The point [tex]\((-8, 19)\)[/tex] is not on the perpendicular bisector.
b. Point [tex]\((1, -8)\)[/tex]
[tex]\[ y = 0.3846153846153846(1) + 7.153846153846154 = 0.3846153846153846 + 7.153846153846154 = 7.538 \][/tex]
[tex]\(-8 ≠ 7.538 \rightarrow The point (1, -8) is not on the perpendicular bisector. c. Point \((0, 19)\)[/tex]
[tex]\[ y = 0.3846153846153846(0) + 7.153846153846154 = 7.153846153846154 \][/tex]
19 ≠ 7.153846153846154 → The point [tex]\((0, 19)\)[/tex] is not on the perpendicular bisector.
d. Point [tex]\((-5, 10)\)[/tex]
[tex]\[ y = 0.3846153846153846(-5) + 7.153846153846154 = -1.923076923076923 + 7.153846153846154 = 5.231 \][/tex]
10 ≠ 5.231 → The point [tex]\((-5, 10)\)[/tex] is not on the perpendicular bisector.
e. Point [tex]\((2, -7)\)[/tex]
[tex]\[ y = 0.3846153846153846(2) + 7.153846153846154 = 0.7692307692307692 + 7.153846153846154 = 7.923 \][/tex]
-7 ≠ 7.923 → The point [tex]\((2, -7)\)[/tex] is not on the perpendicular bisector.
Based on these calculations, none of the given points lie on the perpendicular bisector of the segment connecting [tex]\((-8, 19)\)[/tex] and [tex]\((2, -7)\)[/tex].
