what is the equation for the perpendicular bisector of the line segment whose endpoints are (-7,2) (-1,-6)
bisector is the line that cuts the segment into half and is perpendicular to it. So first we need to find the midpoint M of segment ; x coordinate of M; (-7+-1)/2=-8/2=-4. y coordinate of M; (2+-6)/2= -4/2=-2.So M(-4,-2)
As lines are perpendicular their gradients will be opposite & reciprocal. Lets find the gradient of AB: (2--6)/(-7--1)= 8/-6=-4/3. So gradient of line; 3/4
Equation of line; y=mx+c so substituting coordinates x,y and gradient m we get; -2=3/4(-4)+c solving for c we get; c=1 so line is; y=(3/4)x+1
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, you can follow these steps:
Step 1: Find the midpoint of the line segment.
To find the midpoint of a line segment with endpoints (x1, y1) and (x2, y2), use the midpoint formula:
midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Given endpoints:
Endpoint 1: (-7, 2) with x1 = -7 and y1 = 2
Endpoint 2: (-1, -6) with x2 = -1 and y2 = -6
Midpoint = ((-7 + (-1)) / 2, (2 + (-6)) / 2)
Midpoint = (-8 / 2, -4 / 2)
Midpoint = (-4, -2)
Therefore, the midpoint is (-4, -2).
Step 2: Find the slope of the line segment.
The slope of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
Let's find the slope using the given endpoints:
Slope = (-6 - 2) / (-1 - (-7))
Slope = (-6 - 2) / (-1 + 7)
Slope = (-8) / (6)
Slope = -4/3
So, the slope of the line segment is -4/3.
Step 3: Find the negative reciprocal of the slope.
The negative reciprocal of a slope is obtained by taking the negative of its reciprocal.
Negative reciprocal = -1 / slope
Negative reciprocal = -1 / (-4/3) = 3/4
Step 4: Find the equation of the perpendicular bisector.
The equation of a line in slope-intercept form is: y = mx + b, where m is the slope and b is the y-intercept.
Using the midpoint (-4, -2) and the negative reciprocal of the slope (3/4), we can substitute these values into the equation to find the y-intercept (b).
-2 = (3/4)(-4) + b
-2 = -3 + b
b = -2 + 3
b = 1
Therefore, the y-intercept (b) is 1.
The equation for the perpendicular bisector is:
y = (3/4)x + 1
To find the equation of the perpendicular bisector of a line segment, you can follow these steps:
Step 1: Find the midpoint of the line segment.
The midpoint can be found by using the following formulas:
x-midpoint = (x1 + x2)/2
y-midpoint = (y1 + y2)/2
In this case, the endpoints are (-7, 2) and (-1, -6):
x-midpoint = (-7 + -1)/2 = -8/2 = -4
y-midpoint = (2 + -6)/2 = -4/2 = -2
So, the midpoint is (-4, -2).
Step 2: Find the slope of the line segment.
The slope of the line segment can be found using the formula:
slope = (y2 - y1)/(x2 - x1)
Using the endpoints (-7, 2) and (-1, -6):
slope = (-6 - 2)/(-1 - (-7)) = -8/6 = -4/3
Step 3: Find the negative reciprocal of the slope.
The negative reciprocal of a number is obtained by flipping the fraction and changing the sign of the resulting fraction. In this case, the negative reciprocal of -4/3 is 3/4.
Step 4: Use the negative reciprocal of the slope and the midpoint to find the equation of the perpendicular bisector.
The equation of a line, using the point-slope form, is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope.
Using the point (-4, -2) and the negative reciprocal slope of 3/4:
y - (-2) = 3/4(x - (-4))
y + 2 = 3/4(x + 4)
y + 2 = 3/4x + 3
y = 3/4x + 3 - 2
y = 3/4x + 1
Therefore, the equation of the perpendicular bisector of the line segment with endpoints (-7, 2) and (-1, -6) is y = (3/4)x + 1.