what is the equation for the perpendicular bisector of the line segment whose endpoints are (-7,2) (-1,-6)
To find the equation of the perpendicular bisector of a line segment, we need to determine two things: the midpoint of the line segment and the slope of the line that is perpendicular to it.
Step 1: Finding the Midpoint
To find the midpoint, we can use the formula:
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Given the endpoints:
P₁(-7, 2) and P₂(-1, -6)
Substituting the values into the formula:
Midpoint = ((-7 + -1)/2, (2 + -6)/2)
Midpoint = (-8/2, -4/2)
Midpoint = (-4, -2)
So, the midpoint of the line segment is (-4, -2).
Step 2: Finding the Slope
To find the slope of a line perpendicular to the line segment, we need to determine the negative reciprocal of the slope of the line segment.
The slope of the line segment can be calculated using the formula:
Slope = (y₂ - y₁) / (x₂ - x₁)
Substituting the values:
Slope = (-6 - 2) / (-1 - (-7))
Slope = (-6 - 2) / (-1 + 7)
Slope = -8 / 6
Slope = -4/3
The negative reciprocal of -4/3 is 3/4. So, the slope of the perpendicular line is 3/4.
Step 3: Writing the equation
We now have the midpoint (-4, -2) and the slope of the perpendicular line (3/4). We can use the point-slope form of the equation to write the equation of the line:
y - y₁ = m(x - x₁)
Substituting the values:
y - (-2) = (3/4)(x - (-4))
y + 2 = (3/4)(x + 4)
y + 2 = (3/4)x + 3
y = (3/4)x + 3 - 2
y = (3/4)x + 1
So, the equation of the perpendicular bisector of the line segment with endpoints (-7, 2) and (-1, -6) is y = (3/4)x + 1.
To find the equation of the perpendicular bisector of a line segment, we need to determine the midpoint of the line segment first and find its slope. Then, we find the negative reciprocal of the midpoint's slope to get the slope of the perpendicular bisector. Finally, we use the slope and the midpoint to form the equation using the point-slope form.
Step 1: Find the Midpoint
To find the midpoint of the line segment with endpoints (-7,2) and (-1,-6), we can use the midpoint formula:
Midpoint (h, k) = ((x1 + x2)/2, (y1 + y2)/2)
Let's plug in the values:
Midpoint (h, k) = ((-7 + -1)/2, (2 - 6)/2)
= (-8/2, -4/2)
= (-4, -2)
Therefore, the midpoint of the line segment is (-4, -2).
Step 2: Find the Slope
To find the slope of the line segment with endpoints (-7,2) and (-1,-6), we can use the slope formula:
Slope (m) = (y2 - y1)/(x2 - x1)
Let's plug in the values:
Slope (m) = (-6 - 2)/(-1 - (-7))
= (-8)/6
= -4/3
Therefore, the slope of the line segment is -4/3.
Step 3: Find the Slope of the Perpendicular Bisector
To find the slope of the perpendicular bisector, we take the negative reciprocal of the slope of the line segment. The negative reciprocal of -4/3 is 3/4.
Therefore, the slope of the perpendicular bisector is 3/4.
Step 4: Form the Equation
We now have the slope of the perpendicular bisector (3/4) and the midpoint (-4, -2). Using the point-slope form of an equation:
y - k = m(x - h)
where (h, k) is the midpoint and m is the slope, we can substitute the values and form the equation:
y - (-2) = (3/4)(x - (-4))
y + 2 = (3/4)(x + 4)
y + 2 = (3/4)x + 3
y = (3/4)x + 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.