find two point the perpendicular bisector of line AB. verify your results by showing each point is equidistant from A and B. A(3,0) B (1,4)
slope of AB is -1/2
slope of bisector is 2
midpoint of AB = (2,2)
line is
y-2 = 2(x-2)
now pick any two points on that line and figure the distances needed
To find the perpendicular bisector of line AB, we need to follow these steps:
Step 1: Find the midpoint of line AB
Step 2: Find the slope of line AB
Step 3: Find the negative reciprocal of the slope of line AB (to get the slope of the perpendicular bisector)
Step 4: Use the midpoint and the slope of the perpendicular bisector to find the equation of the line
Step 5: Find the intersection points of the perpendicular bisector with line AB
Step 6: Verify that each point is equidistant from points A and B
Step 1: Find the midpoint of line AB
The midpoint of line AB is the average of the x-coordinates and the average of the y-coordinates of points A and B.
Midpoint(x, y) = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint(x, y) = ((3 + 1)/2, (0 + 4)/2)
Midpoint(x, y) = (2, 2)
So, the midpoint of line AB is (2, 2).
Step 2: Find the slope of line 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)
Slope of line AB = (4 - 0) / (1 - 3)
Slope of line AB = 4 / (-2)
Slope of line AB = -2
Step 3: Find the negative reciprocal of the slope of line AB
To find the slope of the perpendicular bisector, we need to take the negative reciprocal of the slope of line AB. The negative reciprocal of a number is obtained by flipping the fraction and changing the sign.
Slope of perpendicular bisector = -1/(-2)
Slope of perpendicular bisector = 1/2
Step 4: Use the midpoint and the slope of the perpendicular bisector to find the equation of the line
Using the point-slope form of a line:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope, we can substitute the values:
y - 2 = (1/2)(x - 2)
2y - 4 = x - 2
x - 2y = -2
So, the equation of the perpendicular bisector is x - 2y = -2.
Step 5: Find the intersection points of the perpendicular bisector with line AB
To find the intersection points, we solve the system of equations between the perpendicular bisector and line AB. The equation of line AB is given by:
y - y1 = m(x - x1)
Substituting the values:
y - 0 = -2(x - 3)
y = -2x + 6
Solving the system of equations:
x - 2y = -2
-2x + y = 6
Adding the equations:
x + (-2x) - 2y + y = -2 + 6
-x - y = 4
x + y = -4
Multiplying the second equation by -1:
-x - y = 4
x + y = -4
Adding the modified equations:
2x = 0
x = 0
Substituting the value of x into either equation:
0 + y = -4
y = -4
So, the intersection point is (0, -4).
Step 6: Verify that each point is equidistant from points A and B
To verify that each point is equidistant from points A and B, we can calculate the distance between each point and points A and B using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
For point (2, 2):
Distance from (2, 2) to A(3, 0):
Distance = √((3 - 2)^2 + (0 - 2)^2)
Distance = √(1 + 4)
Distance = √5
Distance from (2, 2) to B(1, 4):
Distance = √((1 - 2)^2 + (4 - 2)^2)
Distance = √(1 + 4)
Distance = √5
The distance from (2, 2) to A and B is √5 for both, confirming that (2, 2) is equidistant from A and B.
For point (0, -4):
Distance from (0, -4) to A(3, 0):
Distance = √((3 - 0)^2 + (0 - (-4))^2)
Distance = √(9 + 16)
Distance = √25
Distance = 5
Distance from (0, -4) to B(1, 4):
Distance = √((1 - 0)^2 + (4 - (-4))^2)
Distance = √(1 + 64)
Distance = √65
The distance from (0, -4) to A is 5 and to B is √65, confirming that (0, -4) is equidistant from A and B.