What point is on the perpendicular bisector of the segment with endpoints (3,4) and (9,4)?
To find a point on the perpendicular bisector of a segment with two given endpoints, we can follow these steps:
1. Find the midpoint of the segment.
2. Determine the slope of the segment.
3. Find the negative reciprocal of the slope to determine the slope of the perpendicular bisector.
4. Use the slope and midpoint to derive the equation of the perpendicular bisector.
5. Plug in any value of the x-coordinate into the equation to find the corresponding y-coordinate of a point on the perpendicular bisector.
Let's go through these steps using the given endpoints (3,4) and (9,4):
1. Finding the midpoint:
The midpoint is calculated by averaging the x-coordinates and y-coordinates separately.
Midpoint's x-coordinate = (3 + 9) / 2 = 12 / 2 = 6
Midpoint's y-coordinate = (4 + 4) / 2 = 8 / 2 = 4
Therefore, the midpoint is (6,4).
2. Determining the slope of the segment:
The slope is calculated using the formula: slope = (change in y) / (change in x).
slope = (4 - 4) / (9 - 3) = 0 / 6 = 0
The slope of the segment is 0.
3. Finding the negative reciprocal of the slope:
The negative reciprocal of 0 is undefined. However, for vertical lines with an undefined slope, the slope of the perpendicular bisector is 0.
4. Deriving the equation of the perpendicular bisector:
Since the slope of the perpendicular bisector is 0, the equation will be in the form y = c, where c is the y-coordinate of the midpoint.
Therefore, the equation of the perpendicular bisector is y = 4.
5. Finding a point on the perpendicular bisector:
We can pick any value for the x-coordinate and substitute it into the equation to solve for y. Let's choose x = 0.
Plugging x = 0 into the equation y = 4, we get:
y = 4
Hence, a point on the perpendicular bisector is (0, 4).
Therefore, the point (0,4) is on the perpendicular bisector of the segment with endpoints (3,4) and (9,4).
To find the point on the perpendicular bisector of a segment, we first need to find the midpoint of the segment, and then determine the slope of the perpendicular bisector.
The midpoint of the segment with endpoints (3, 4) and (9, 4) can be found by using the midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Substituting the coordinates, we get:
Midpoint = ((3 + 9) / 2, (4 + 4) / 2)
= (12 / 2, 8 / 2)
= (6, 4)
Thus, the midpoint of the segment is (6, 4).
To find the slope of the perpendicular bisector, we need to find the negative reciprocal of the slope of the original segment. The slope of the original segment can be found using the slope formula:
Slope = (y2 - y1) / (x2 - x1)
Substituting the coordinates, we get:
Slope = (4 - 4) / (9 - 3)
= 0 / 6
= 0
Since the slope is 0, the perpendicular bisector of the segment will have an undefined slope.
Therefore, the equation of the perpendicular bisector will be in the form x = k, where k is the x-coordinate of the midpoint.
In this case, the equation of the perpendicular bisector is x = 6.
Thus, any point on the perpendicular bisector will have an x-coordinate of 6.
Wouldn't the midpoint be one of those points?
So find the midpoint of the given segment.