what is the equation for the perpendicular bisector of the line segment whose endpoints are (-7,2) (-1,-6)
Indicate the equation of the line that is the perpendicular bisector of the segment with endpoints (4, 1) and (2, -5).
To find the equation for the perpendicular bisector of a line segment, we need to find the midpoint of the line segment and the slope of the line containing the segment.
Step 1: Find the midpoint of the line segment
To find the midpoint, we use the formula:
midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Given endpoints: (-7, 2) and (-1, -6)
midpoint = ((-7 + (-1))/2, (2 + (-6))/2)
midpoint = (-8/2, -4/2)
midpoint = (-4, -2)
The midpoint of the line segment is (-4,-2).
Step 2: Find the slope of the line segment
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
Given endpoints: (-7, 2) and (-1, -6)
slope = (-6 - 2) / (-1 - (-7))
slope = (-8) / (6)
slope = -4/3
The slope of the line segment is -4/3.
Step 3: Find the negative reciprocal of the slope
To find the slope of the perpendicular bisector, we need to flip the sign of the slope and take its reciprocal:
perpendicular bisector slope = -1 / (-4/3)
perpendicular bisector slope = 3/4
Step 4: Use the midpoint and the perpendicular bisector slope to find the equation of the line
Using the slope-intercept form of a line, which is y = mx + b, we can substitute the slope and the midpoint to find the equation of the perpendicular bisector.
(-4, -2) is a point on the line, so we can substitute these values in the equation:
y = (3/4)x + b
Substituting (-4, -2):
-2 = (3/4)(-4) + b
-2 = -3 + b
b = -2 + 3
b = 1
Therefore, the equation of the perpendicular bisector of the line segment is:
y = (3/4)x + 1
To find the equation of the perpendicular bisector of a line segment, we need to find its slope and midpoint.
Step 1: Find the midpoint of the line segment:
The midpoint is calculated by finding the average of the x-coordinates and the average of the y-coordinates of the two endpoints.
Midpoint formula:
(x1 + x2)/2, (y1 + y2)/2
In this case:
Midpoint = (-7 + -1)/2, (2 + -6)/2
= (-8)/2, (-4)/2
= -4, -2
So, the midpoint of the line segment is (-4, -2).
Step 2: Find the slope of the line segment:
The slope of a line can be calculated using the formula:
slope = (y2 - y1)/(x2 - x1)
In this case:
Slope = (-6 - 2)/(-1 - (-7))
= (-8)/(6)
= -4/3
Step 3: Find the negative reciprocal of the slope:
To find the slope of a line perpendicular to another line, we need to take the negative reciprocal of its slope. In other words, we flip the fraction and change the sign.
Negative reciprocal of -4/3 = 3/4
Step 4: Use the midpoint and the negative reciprocal of the slope to write the equation:
Since we have the midpoint (-4, -2) and the slope 3/4, we can use the point-slope form of a line to write the equation:
y - y1 = m(x - x1)
Using the values:
y - (-2) = 3/4(x - (-4))
Simplifying:
y + 2 = 3/4(x + 4)
Multiplying both sides by 4 to get rid of the fraction:
4y + 8 = 3(x + 4)
Distributing 3 to the terms inside the parentheses:
4y + 8 = 3x + 12
Moving all terms to the left side to get the equation in standard form:
3x - 4y = 4
Therefore, the equation for the perpendicular bisector of the line segment with endpoints (-7, 2) and (-1, -6) is 3x - 4y = 4.
The formula is y-y1= m*(x-x1)
1. Find the midpoint= [(x1-x2)/2, (y1-y2)/2]
you will get (-4,-2)
2. Find slope m1= [2-(-6)]/[-7-(-1)]= 8/-6=-4/3
3. find the slope of the perpendicular line
m1*m2=-1
-4/3*m2= -1
m2= 3/4
4. Plug in given points (-7,2) and (-1,-6) and m2 in the formula that I wrote in the beginning
y-(-2)=3/4(x-(-4))
y+2=3/4(x+4)
y+2=3/4x+3
y=3/4x+1 this is the equation your were looking for.
Please, make sure the numbers are correct and double check the math (in case of typo), but the actual idea is correct.