Indicate the equation of the line that is the perpendicular bisector of the segment with endpoints (4, 1) and (2, -5).
midpoint is ( (4+2)/2 , (1-5)/2 ) = (3,-2)
slope for given points line = (1+5)/(4-2) = 3
so slope of perpendicular is -1/3
so y = (-1/3)x + b
sub in (3,-2)
-2 = (-1/3)(3) + b
b = -1
equation : y = (-1/3)x -1
To find the equation of the line that is the perpendicular bisector of a segment, we need to find the midpoint of the segment and then determine the slope of the line.
Step 1: Find the midpoint of the segment.
The midpoint of a segment with endpoints (x₁, y₁) and (x₂, y₂) can be found using the formula:
(midpoint_x, midpoint_y) = ((x₁ + x₂)/2, (y₁ + y₂)/2)
In this case, the endpoints are (4, 1) and (2, -5). So the midpoint is:
(midpoint_x, midpoint_y) = ((4 + 2)/2, (1 + (-5))/2)
= (3, -2)
Step 2: Determine the slope of the line.
The slope of the line perpendicular to a given line is the negative reciprocal of the slope of the given line. So, we need to find the slope of the line that passes through the given endpoints.
The formula for slope is:
slope = (y₂ - y₁)/(x₂ - x₁)
Using the endpoints (4, 1) and (2, -5), the slope of the line is:
slope = (-5 - 1)/(2 - 4)
= (-6)/(-2)
= 3
So, the slope of the line perpendicular to the line passing through the endpoints is the negative reciprocal of 3, which is -1/3.
Step 3: Find the equation of the line.
The equation of a line with slope m and passing through the point (x₁, y₁) can be expressed in slope-intercept form: y = mx + b, where b is the y-intercept.
Substituting the midpoint (3, -2) into the equation, we have:
-2 = (-1/3)(3) + b
-2 = -1 + b
Solving for b:
b = -2 + 1
b = -1
Therefore, the equation of the line that is the perpendicular bisector of the segment with endpoints (4, 1) and (2, -5) is:
y = (-1/3)x - 1
To find the equation of the line that is the perpendicular bisector of a segment, we can follow these steps:
1. Find the midpoint of the segment.
2. Determine the slope of the given segment.
3. Find the negative reciprocal of the slope to obtain the slope of the perpendicular bisector.
4. Use the midpoint and the slope of the perpendicular bisector to write the equation in point-slope form.
5. Simplify the equation to get the final answer.
Let's go through each step:
1. Finding the midpoint:
The midpoint formula is given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Substituting the given coordinates (4, 1) and (2, -5):
Midpoint = ((4 + 2)/2, (1 + -5)/2)
= (6/2, -4/2)
= (3, -2)
2. Determining the slope of the given segment:
The slope formula is given by:
Slope = (y2 - y1)/(x2 - x1)
Substituting the given coordinates (4, 1) and (2, -5):
Slope = (-5 - 1)/(2 - 4)
= (-6)/(-2)
= 3
3. Finding the negative reciprocal of the slope:
The negative reciprocal of a number 'm' is represented as -1/m.
Therefore, the negative reciprocal of 3 is -1/3.
4. Writing the equation in point-slope form:
The point-slope form of a line is given by:
y - y1 = m(x - x1)
Substituting the midpoint (3, -2) and the slope -1/3:
y - (-2) = (-1/3)(x - 3)
y + 2 = (-1/3)(x - 3)
5. Simplifying the equation:
Multiply both sides of the equation by -3 to eliminate the fraction:
-3(y + 2) = (-1)(x - 3)
-3y - 6 = -x + 3
-3y = -x + 9
Finally, rearrange the equation to the standard form (Ax + By + C = 0):
x + 3y - 9 = 0
Therefore, the equation of the line that is the perpendicular bisector of the segment with endpoints (4, 1) and (2, -5) is x + 3y - 9 = 0.