Find the equation of the perpendicular line passing through the midpoint of the line segment connecting
(−2, 7)
and
(7, −1).
steps:
- find the midpoint for the given points
- find the slope of the line passing through those points
- the slope of the line you want is the negative reciprocal of that slope
- find the equation using the method you learned.
To find the equation of the perpendicular line passing through the midpoint of the line segment, we need to follow these steps:
1. Find the midpoint of the line segment:
To find the midpoint, we can use the midpoint formula, which states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are given by:
Midpoint(x, y) = ((x1 + x2) / 2, (y1 + y2) / 2)
Let's substitute the coordinates of the endpoints given in the question:
Midpoint(x, y) = ((-2 + 7) / 2, (7 + (-1)) / 2)
Midpoint(x, y) = (5 / 2, 6 / 2)
Midpoint(x, y) = (5 / 2, 3)
So the midpoint of the line segment is (5 / 2, 3).
2. Find the slope of the given line segment:
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
Slope = (y2 - y1) / (x2 - x1)
Let's substitute the coordinates of the endpoints given in the question:
Slope = (-1 - 7) / (7 - (-2))
Slope = (-8) / (9)
Slope = -8 / 9
3. To find the slope of the perpendicular line, we need to use the property that the slopes of two perpendicular lines are negative reciprocals of each other. So, the slope of the perpendicular line is 9 / 8.
4. Using the slope-intercept form of a line, which is given by y = mx + b, where m is the slope and b is the y-intercept, we can find the equation of the perpendicular line.
We know the slope (9 / 8) and we need to find the y-intercept (b).
Let's substitute the values of the midpoint (5 / 2, 3) into the equation:
3 = (9 / 8) * (5 / 2) + b
3 = 45 / 16 + b
b = 48 / 16 - 45 / 16
b = 3 / 16
So the equation of the perpendicular line passing through the midpoint is y = (9 / 8)x + (3 / 16).