Final answer:
To prove the line is perpendicular, we found the midpoint of the line segment, calculated the slope of the segment, and verified that the slope of our line is the negative reciprocal of it. The calculations confirm that the line is indeed perpendicular to the segment.
Step-by-step explanation:
To show that the line passing through the point (3,4) and the midpoint of the line segment joining the points (-1,1) and (3,9) is perpendicular to the line segment, we need to first find the midpoint of the line segment and then ensure that the slope of our line is the negative reciprocal of the line segment's slope.
Step 1: Finding the midpoint
The midpoint (M) of a line segment with endpoints (x1, y1) and (x2, y2) is given by:
M = ((x1 + x2)/2, (y1 + y2)/2)
So, for the line segment joining (-1,1) and (3,9):
M = ((-1 + 3)/2, (1 + 9)/2) = (1, 5)
Step 2: Determining the slope of the line segment
The slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
For our line segment, m = (9 - 1) / (3 - (-1)) = 8/4 = 2
Step 3: Ensuring perpendicularity
For two lines to be perpendicular, the slope of one line must be the negative reciprocal of the other. Since the slope of the line segment is 2, the slope of our line must be -1/2.
The line through (3,4) and (1,5) then has slope:
slope = (5 - 4) / (1 - 3) = 1 / (-2) = -1/2
Since this slope is the negative reciprocal of the line segment's slope, we can confirm that the line through (3,4) and the midpoint (1,5) is perpendicular to the line segment joining (-1,1) and (3,9).