What part of the coordinate plane is equidistant from the points A(–3,2) and B(3,2)?
Since point A and B have the same y-coordinate, any point on the line y = 2 will be equidistant from A and B. Thus, the part of the coordinate plane that is equidistant from A and B is the line y = 2.
To find the part of the coordinate plane that is equidistant from points A(-3,2) and B(3,2), we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint between two points (x₁, y₁) and (x₂, y₂) are given by the following equations:
x = (x₁ + x₂) / 2
y = (y₁ + y₂) / 2
In this case, the points are A(-3,2) and B(3,2). Let's plug in the values into the formula:
x = (-3 + 3) / 2 = 0 / 2 = 0
y = (2 + 2) / 2 = 4 / 2 = 2
Therefore, the point (0,2) is equidistant from both A(-3,2) and B(3,2) in the coordinate plane.
To determine the part of the coordinate plane that is equidistant from points A(-3, 2) and B(3, 2), we need to find the perpendicular bisector of line segment AB.
The perpendicular bisector is a line that passes through the midpoint of AB and intersects it at a right angle. Since the y-coordinates of A and B are the same (both 2), the midpoint of AB lies on the line y = 2.
To find the x-coordinate of the midpoint, we take the average of the x-coordinates of A and B: (-3 + 3)/2 = 0/2 = 0. So, the midpoint is M(0, 2).
Next, we find the slope of AB. The slope formula is given by:
m = (y₂ - y₁) / (x₂ - x₁)
Substituting the coordinates of A and B into the formula, we have:
m = (2 - 2) / (3 - (-3)) = 0 / 6 = 0
Since the slope of AB is 0, the slope of the perpendicular bisector will be the negative reciprocal of 0, which is undefined. Therefore, the equation of the perpendicular bisector is x = 0.
So, the part of the coordinate plane equidistant from points A and B is the vertical line x = 0, which coincides with the y-axis.