Which equation represents the perpendicular bisector of AB whose endpoints are A(4, 1) andB(0,3)
no choices, but it will have to go through (2,2) with slope 2
Why did the line cross the Cartesian plane? Because it couldn't resist the temptation to divide AB into two equal parts! The equation representing the perpendicular bisector of AB is y = -x + 5.
To find the equation of the perpendicular bisector of line AB, we need to determine its slope and midpoint.
Step 1: Find the midpoint of AB:
The midpoint of a line segment with endpoints A(x₁, y₁) and B(x₂, y₂) is given by the coordinates ((x₁ + x₂)/2, (y₁ + y₂)/2).
In this case, the coordinates of A are (4, 1) and the coordinates of B are (0, 3).
Using the midpoint formula, we can calculate the midpoint:
Midpoint = ((4 + 0)/2, (1 + 3)/2)
= (4/2, 4/2)
= (2, 2)
The midpoint of AB is (2, 2).
Step 2: Find the slope of AB:
The slope of a line with two points A(x₁, y₁) and B(x₂, y₂) is given by (y₂ - y₁) / (x₂ - x₁).
Let's use the coordinates (4, 1) for point A and (0, 3) for point B to calculate the slope:
Slope of AB = (3 - 1) / (0 - 4)
= 2 / -4
= -1/2
The slope of AB is -1/2.
Step 3: Find the negative reciprocal of the slope:
To find the slope of the perpendicular line, we need to take the negative reciprocal of the slope of AB.
Negative reciprocal of -1/2 is 2.
Step 4: Use the midpoint and slope to write the equation:
Now we have the midpoint (2, 2) and the slope of the perpendicular bisector, which is 2.
Using the point-slope form of a linear equation, we have:
y - y₁ = m(x - x₁)
Substituting the values:
y - 2 = 2(x - 2)
Simplifying:
y - 2 = 2x - 4
Rearranging the equation into the slope-intercept form (y = mx + b):
y = 2x - 4 + 2
y = 2x - 2
Therefore, the equation of the perpendicular bisector of AB is y = 2x - 2.
To find the equation of the perpendicular bisector of AB, we can start by finding the midpoint of AB, and then determine the slope of the line AB. From there, we can find the negative reciprocal of the slope to get the slope of the perpendicular bisector. Finally, using the midpoint and the slope of the perpendicular bisector, we can write the equation in point-slope form.
Step 1: Find the midpoint of AB.
The midpoint (x, y) of two points A(x₁, y₁) and B(x₂, y₂) is given by the formulas:
x = (x₁ + x₂) / 2
y = (y₁ + y₂) / 2
In this case, A(4, 1) and B(0, 3), so the midpoint is:
x = (4 + 0) / 2 = 2
y = (1 + 3) / 2 = 2
The midpoint of AB is M(2, 2).
Step 2: Find the slope of AB.
The slope (m) of a line passing through two points A(x₁, y₁) and B(x₂, y₂) is given by the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Using the coordinates of A and B, we have:
m = (3 - 1) / (0 - 4)
= 2 / -4
= -1/2
So, the slope of AB is -1/2.
Step 3: Find the slope of the perpendicular bisector.
The slope of the perpendicular bisector is the negative reciprocal of the slope of AB. To find it, we take the negative reciprocal of -1/2, which means we flip the fraction and change its sign:
Slope of the perpendicular bisector = -1 / (-1/2)
= 2
So, the slope of the perpendicular bisector is 2.
Step 4: Write the equation of the perpendicular bisector in point-slope form.
The point-slope form of a linear equation is given by:
y - y₁ = m(x - x₁)
Using the midpoint M(2, 2) and the slope 2 (from Step 3), we can write the equation:
y - 2 = 2(x - 2)
Expanding and simplifying the equation further:
y - 2 = 2x - 4
y = 2x - 2
Therefore, the equation representing the perpendicular bisector of AB is y = 2x - 2.