Answer :
To determine which points lie on the perpendicular bisector of the given segment, we need to go through a series of steps, including calculating the midpoint of the segment, finding the slope of the segment and then the slope of the perpendicular bisector. Finally, we will check which given points satisfy the equation of the perpendicular bisector.
Here are the detailed steps:
1. Identify the Coordinates of the Segment Endpoints:
- Point 1: [tex]\((-8, 19)\)[/tex]
- Point 2: [tex]\((2, -7)\)[/tex]
2. Calculate the Midpoint of the Segment:
The midpoint [tex]\((M_x, M_y)\)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be found using the formula:
[tex]\[ M_x = \frac{x_1 + x_2}{2}, \quad M_y = \frac{y_1 + y_2}{2} \][/tex]
Plugging in the given points:
[tex]\[ M_x = \frac{-8 + 2}{2} = \frac{-6}{2} = -3 \][/tex]
[tex]\[ M_y = \frac{19 + (-7)}{2} = \frac{12}{2} = 6 \][/tex]
Therefore, the midpoint is [tex]\((-3, 6)\)[/tex].
3. Calculate the Slope of the Original Segment:
The slope [tex]\(m\)[/tex] of a segment with endpoints [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]
Using the given points:
[tex]\[ m = \frac{-7 - 19}{2 - (-8)} = \frac{-26}{10} = -2.6 \][/tex]
4. Determine the Slope of the Perpendicular Bisector:
The slope of the perpendicular bisector of a segment is the negative reciprocal of the original slope. If the slope of the original segment is [tex]\(m\)[/tex], the slope of the perpendicular bisector is [tex]\(-\frac{1}{m}\)[/tex]:
[tex]\[ \text{Slope of perpendicular bisector} = -\frac{1}{-2.6} = 0.3846153846153846 \][/tex]
5. Equation of the Perpendicular Bisector:
Using point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
[tex]\[ y - 6 = 0.3846153846153846 (x + 3) \][/tex]
6. Check which Points Lie on the Perpendicular Bisector:
We need to verify if the given points satisfy the equation [tex]\( y - 6 = 0.3846153846153846 (x + 3) \)[/tex]:
- [tex]\((-8, 19)\)[/tex]:
[tex]\[ 19 - 6 \neq 0.3846153846153846 (-8 + 3) \quad \text{(13 ≠ -1.92)} \][/tex]
- [tex]\((1, -8)\)[/tex]:
[tex]\[ -8 - 6 \neq 0.3846153846153846 (1 + 3) \quad \text{(-14 ≠ 1.538)} \][/tex]
- [tex]\((0, 19)\)[/tex]:
[tex]\[ 19 - 6 \neq 0.3846153846153846 (0 + 3) \quad \text{(13 ≠ 1.154)} \][/tex]
- [tex]\((-5, 10)\)[/tex]:
[tex]\[ 10 - 6 \neq 0.3846153846153846 (-5 + 3) \quad \text{(4 ≠ 0.769)} \][/tex]
- [tex]\((2, -7)\)[/tex]:
[tex]\[ -7 - 6 \neq 0.3846153846153846 (2 + 3) \quad \text{(-13 ≠ 1.923)} \][/tex]
From these checks, none of the points lie on the perpendicular bisector; hence the output is an empty set:
The points on the perpendicular bisector are:
[tex]\[ [] \][/tex]
Here are the detailed steps:
1. Identify the Coordinates of the Segment Endpoints:
- Point 1: [tex]\((-8, 19)\)[/tex]
- Point 2: [tex]\((2, -7)\)[/tex]
2. Calculate the Midpoint of the Segment:
The midpoint [tex]\((M_x, M_y)\)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be found using the formula:
[tex]\[ M_x = \frac{x_1 + x_2}{2}, \quad M_y = \frac{y_1 + y_2}{2} \][/tex]
Plugging in the given points:
[tex]\[ M_x = \frac{-8 + 2}{2} = \frac{-6}{2} = -3 \][/tex]
[tex]\[ M_y = \frac{19 + (-7)}{2} = \frac{12}{2} = 6 \][/tex]
Therefore, the midpoint is [tex]\((-3, 6)\)[/tex].
3. Calculate the Slope of the Original Segment:
The slope [tex]\(m\)[/tex] of a segment with endpoints [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]
Using the given points:
[tex]\[ m = \frac{-7 - 19}{2 - (-8)} = \frac{-26}{10} = -2.6 \][/tex]
4. Determine the Slope of the Perpendicular Bisector:
The slope of the perpendicular bisector of a segment is the negative reciprocal of the original slope. If the slope of the original segment is [tex]\(m\)[/tex], the slope of the perpendicular bisector is [tex]\(-\frac{1}{m}\)[/tex]:
[tex]\[ \text{Slope of perpendicular bisector} = -\frac{1}{-2.6} = 0.3846153846153846 \][/tex]
5. Equation of the Perpendicular Bisector:
Using point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
[tex]\[ y - 6 = 0.3846153846153846 (x + 3) \][/tex]
6. Check which Points Lie on the Perpendicular Bisector:
We need to verify if the given points satisfy the equation [tex]\( y - 6 = 0.3846153846153846 (x + 3) \)[/tex]:
- [tex]\((-8, 19)\)[/tex]:
[tex]\[ 19 - 6 \neq 0.3846153846153846 (-8 + 3) \quad \text{(13 ≠ -1.92)} \][/tex]
- [tex]\((1, -8)\)[/tex]:
[tex]\[ -8 - 6 \neq 0.3846153846153846 (1 + 3) \quad \text{(-14 ≠ 1.538)} \][/tex]
- [tex]\((0, 19)\)[/tex]:
[tex]\[ 19 - 6 \neq 0.3846153846153846 (0 + 3) \quad \text{(13 ≠ 1.154)} \][/tex]
- [tex]\((-5, 10)\)[/tex]:
[tex]\[ 10 - 6 \neq 0.3846153846153846 (-5 + 3) \quad \text{(4 ≠ 0.769)} \][/tex]
- [tex]\((2, -7)\)[/tex]:
[tex]\[ -7 - 6 \neq 0.3846153846153846 (2 + 3) \quad \text{(-13 ≠ 1.923)} \][/tex]
From these checks, none of the points lie on the perpendicular bisector; hence the output is an empty set:
The points on the perpendicular bisector are:
[tex]\[ [] \][/tex]
