Answer:
1, the mid point
The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) can be found using the midpoint formula:
midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Substituting the given endpoints (4, -4) and (0, 9), we get:
midpoint = ((4 + 0) / 2, (-4 + 9) / 2) = (2, 2.5)
Therefore, the midpoint of the line segment with endpoints (4, -4) and (0, 9) is (2, 2.5).
2, the slope
The slope of the line passing through the points (4, -4) and (0, 9) can be found using the formula:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Substituting the given coordinates, we get:
slope = (9 - (-4)) / (0 - 4) = 13 / (-4) = -3.25
Therefore, the slope of the line is -3.25.
3, the length
The length of the line passing through the points (4, -4) and (0, 9) can be found using the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the given coordinates, we get:
distance = sqrt((0 - 4)^2 + (9 - (-4))^2) = sqrt(16 + 169) = sqrt(185)
Therefore, the length of the line is sqrt(185).
4, the equation
The equation of the line passing through the points (4, -4) and (0, 9) can be found using the point-slope form of the equation of a line:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is one of the points.
Substituting the given slope and one of the points, we get:
y - (-4) = -3.25(x - 4)
Simplifying, we get:
y + 4 = -3.25x + 13
y = -3.25x + 9
Therefore, the equation of the line passing through the points (4, -4) and (0, 9) is y = -3.25x + 9.