Find 2 points on the perpendicular bisector of line AB. Verify your results by showing each point is equidistant from A and B.
34.) A (0,0) B 90,4)
36.) A (3,3) B (3,-3)
38.) A (3,0) B (1,4)
36 is easy, since AB is vertical.
34 and 38 are done the same way:
find the midpoint of AB
find the slope of AB; you want the negative reciprocal.
Now you have a point and a slope; the equation is easy
#38:
midpoint of AB is (2,2)
slope of AB is -2, so slope of ┴ is 1/2
y-2 = 1/2 (x-2)
now just pick any two values of x or y and solve for the other (y or x)
To find the perpendicular bisector of line AB, we need to first calculate the midpoint of line AB. The midpoint is the average of the x-coordinates and the average of the y-coordinates of points A and B.
34.) For A(0,0) and B(90,4):
The midpoint is ((0 + 90)/2, (0 + 4)/2) = (45, 2).
The slope of line AB is (4 - 0)/(90 - 0) = 4/90 = 2/45.
The negative reciprocal of the slope is -1/(2/45) = -45/2.
Using the point-slope formula, we have:
y - 2 = (-45/2)(x - 45).
Simplifying the equation:
y - 2 = -45x/2 + (45 * 45 * 45)/2.
y - 2 = -45x/2 + 10125/2.
y = -45x/2 + 10125/2 + 2.
y = -45x/2 + 10129/2.
Now we can find two points on the perpendicular bisector by substituting values for x:
Let's choose x = 0:
y = -45(0)/2 + 10129/2 = 10129/2 = 5064.5.
The first point is (0, 5064.5).
Let's choose x = 45:
y = -45(45)/2 + 10129/2 = -2025/2 + 5064.5 = 3039.5.
The second point is (45, 3039.5).
To verify that these points are equidistant from A and B, we can calculate the distance between each point and points A and B.
Distance between (0, 5064.5) and A(0,0):
√((0 - 0)^2 + (5064.5 - 0)^2) = √(0 + 25649400.25) = √25649400.25 = 5064.5.
Distance between (0, 5064.5) and B(90,4):
√((90 - 0)^2 + (4 - 5064.5)^2) = √(8100 + 25650810.25) = √25658910.25 = 5064.5.
Distance between (45, 3039.5) and A(0,0):
√((0 - 45)^2 + (3039.5 - 0)^2) = √(2025 + 9237102.25) = √9239127.25 ≈ 3039.5.
Distance between (45, 3039.5) and B(90,4):
√((90 - 45)^2 + (4 - 3039.5)^2) = √(2025 + 9223435.25) = √9225450.25 ≈ 3039.5.
As we can see, both points are equidistant from points A and B.
36.) For A(3,3) and B(3,-3):
The midpoint is ((3 + 3)/2, (3 + -3)/2) = (6/2, 0) = (3,0).
The slope of line AB is (-3 - 3)/(3 - 3) = -6/0, which is undefined as it represents a vertical line.
The perpendicular bisector will be a horizontal line passing through the midpoint (3,0).
Any point on the line will have a y-coordinate of 0.
Two points on the perpendicular bisector are (3,0) and (x,0).
Since the line is horizontal, the x-coordinate of the second point can be any value.
To verify that these points are equidistant from A and B, we can calculate the distance between each point and points A and B.
Distance between (3, 0) and A(3,3):
√((3 - 3)^2 + (0 - 3)^2) = √(0 + 9) = √9 = 3.
Distance between (3, 0) and B(3,-3):
√((3 - 3)^2 + (0 - -3)^2) = √(0 + 9) = √9 = 3.
Since both points are equidistant from points A and B, we do not need to find another point on the perpendicular bisector.
38.) For A(3,0) and B(1,4):
The midpoint is ((3 + 1)/2, (0 + 4)/2) = (2, 2).
The slope of line AB is (4 - 0)/(1 - 3) = 4/-2 = -2.
The negative reciprocal of the slope is -1/(-2) = 1/2.
Using the point-slope formula, we have:
y - 2 = (1/2)(x - 2).
Simplifying the equation:
y - 2 = (1/2)x - 1.
y = (1/2)x + 1.
Now we can find two points on the perpendicular bisector by substituting values for x:
Let's choose x = 0:
y = (1/2)(0) + 1 = 0 + 1 = 1.
The first point is (0, 1).
Let's choose x = 4:
y = (1/2)(4) + 1 = 2 + 1 = 3.
The second point is (4, 3).
To verify that these points are equidistant from A and B, we can calculate the distance between each point and points A and B.
Distance between (0, 1) and A(3,0):
√((3 - 0)^2 + (0 - 1)^2) = √(9 + 1) = √10.
Distance between (0, 1) and B(1,4):
√((1 - 0)^2 + (4 - 1)^2) = √(1 + 9) = √10.
Distance between (4, 3) and A(3,0):
√((3 - 4)^2 + (0 - 3)^2) = √(1 + 9) = √10.
Distance between (4, 3) and B(1,4):
√((1 - 4)^2 + (4 - 3)^2) = √(9 + 1) = √10.
As we can see, both points are equidistant from points A and B.