Answer:
a) the midpoint is (1.5, 2.5)
b) the line is y = -(7/3)*x + 6.
Explanation:
a)
Suppose we have two values, A and B, the mid-value between A and B is:
(A + B)/2
Now, if we have a segment with endpoints (a, b) and (c, d), the midpoint will be in the mid-value of the x-components and the mid-value of the y-components, this means that the midpoint is:
( (c + a)/2, (b + d)/2)
a) Then if the endpoints of the segment are (-2, 1) and (5, 4), the midpoint of this segment will be:
( (-2 + 5)/2, (1 + 4)/2) = (3/2, 5/2) = (1.5, 2,5)
The midpoint of the segment is (1.5, 2.5)
b)
Now we want to find the equation of a perpendicular line to our segment, that passes through the point (1.5, 2.5).
First, if we have a line:
y = a*x + b
A perpendicular line to this one will have a slope equal to -(1/a)
So the first thing we need to do is find the slope of the graphed segment.
We know that for a line that passes through the points (a, b) and (c, d) the slope is:
slope = (c - a)/(d - b)
Then the slope of the segment is:
slope = (4 - 1)/(5 - (-2)) = 3/7
Then the slope of the perpendicular line will be:
s = -(7/3)
Then the perpendicular line will be something like:
y = -(7/3)*x + d
Now we want this line to pass through the point (1.5, 2.5), then we can replace the values of this point in the above equation, and solve for d.
2.5 = -(7/3)*1.5 + d
2.5 + (7/3)*1.5 = d = 6
Then the line is:
y = -(7/3)*x + 6