Find the equation of the perpendicular bisector of the straight line joining the points A(-1,0)and B(3,4).
find the midpoint: (1,2)
find the slope: 1
slope of perp. line: -1
so, now you have a point and a slope:
y-2 = -(x-1)
To find the equation of the perpendicular bisector of the line joining the points A(-1,0) and B(3,4), follow these steps:
Step 1: Find the midpoint of the line segment AB.
The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by the following formulas:
Midpoint_x = (x₁ + x₂) / 2
Midpoint_y = (y₁ + y₂) / 2
Applying the formulas with A(-1, 0) and B(3, 4):
Midpoint_x = (-1 + 3) / 2 = 2 / 2 = 1
Midpoint_y = (0 + 4) / 2 = 4 / 2 = 2
Therefore, the midpoint M of line segment AB is (1, 2).
Step 2: Find the slope of the line segment AB.
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
Slope = (y₂ - y₁) / (x₂ - x₁)
Applying the formula with A(-1, 0) and B(3, 4):
Slope = (4 - 0) / (3 - (-1)) = 4 / 4 = 1
Therefore, the slope of line segment AB is 1.
Step 3: Find the negative reciprocal of the slope.
The negative reciprocal of a number is obtained by changing its sign and taking the reciprocal (flipping the fraction).
Since the slope of AB is 1, the negative reciprocal is -1.
Step 4: Use the midpoint and the negative reciprocal of the slope to find the equation of the perpendicular bisector.
The equation of a line with slope m passing through the point (x₁, y₁) can be written in the point-slope form as:
y - y₁ = m(x - x₁)
Substituting the values into the formula with the midpoint (1, 2) and the negative reciprocal of the slope (-1):
y - 2 = -1(x - 1)
Simplifying the equation:
y - 2 = -x + 1
y = -x + 3
Therefore, the equation of the perpendicular bisector of the line joining points A(-1, 0) and B(3, 4) is y = -x + 3.
To find the equation of the perpendicular bisector of the line joining the points A(-1, 0) and B(3, 4), we can follow these steps:
Step 1: Find the midpoint of the line segment AB.
Step 2: Find the slope of the line AB.
Step 3: Determine the negative reciprocal (perpendicular) slope of the line AB.
Step 4: Use the midpoint and the perpendicular slope to find the equation of the perpendicular bisector.
Let's go through these steps:
Step 1: Find the midpoint of the line segment AB.
The midpoint formula is given by:
Midpoint = [ (x₁ + x₂)/2 , (y₁ + y₂)/2 ]
Using the coordinates of points A(-1, 0) and B(3, 4), substitute these values into the midpoint formula:
Midpoint = [ ( -1 + 3 )/2 , ( 0 + 4 )/2 ]
= [ 2/2 , 4/2 ]
= [ 1, 2 ]
So the midpoint of line AB is (1, 2).
Step 2: Find the slope of the line AB.
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by:
Slope = ( Δy / Δx )
= ( y₂ - y₁ ) / ( x₂ - x₁ )
Using the coordinates of points A(-1, 0) and B(3, 4), substitute these values into the slope formula:
Slope = ( 4 - 0 ) / ( 3 - (-1) )
= 4 / 4
= 1
So the slope of line AB is 1.
Step 3: Determine the negative reciprocal (perpendicular) slope of the line AB.
The negative reciprocal of a line with slope m is given by:
Perpendicular slope = -1 / m
In this case, the slope of line AB is 1.
Perpendicular slope = -1 / 1
= -1
So the negative reciprocal (perpendicular) slope of line AB is -1.
Step 4: Use the midpoint and the perpendicular slope to find the equation of the perpendicular bisector.
We can use the point-slope form of a line to find the equation of the perpendicular bisector, which is given by:
y - y₁ = m * (x - x₁)
Substitute the midpoint (x₁, y₁) = (1, 2) and the perpendicular slope m = -1 into the point-slope form:
y - 2 = -1 * (x - 1)
Now, we can simplify and rearrange the equation:
y - 2 = -x + 1
Add x to both sides:
y + x - 1 = 2
Rearrange the equation to the standard form:
x + y = 3
Therefore, the equation of the perpendicular bisector of the line joining points A(-1, 0) and B(3, 4) is x + y = 3.