(1) What is the slope of the line containing the midpoint of the segment with endpoints at (2, 4) and (0, -2) and the midpoint of the segment with endpoints at (5, 1) and (1, 5)? Express your answer in simplest form.
(2) The lines -2x + y = k and 0.5x + y = 14 intersect when x = -8.4. What is the value of k?
Thank you!
1) To find the x of the midpoint, add the x values of the endpoints, then divide by 2
To find the y of the midpoint, add the yx values of the endpoints, then divide by 2
for the 1st segment, the x of the midpoint is (-2+0)/2 = -1
the y of the midpoint is (-2+4)/2 = 1
So the midpoint of the first segment is (-1,1)
now find the midpoint of the second segment in the same way.
Now that you have the 2 points, find the slope (assuming you know how to do that)
2. sub x = -8.4 into 0.5x + y = 14 and solve for y
You now have the intersection point.
sub that point into the first equation to find k
(1) To find the slope of a line given two points, you can use the formula:
slope = (change in y-coordinates)/(change in x-coordinates)
Let's find the coordinates of the midpoint for the first segment:
x-coordinate of midpoint = (x-coordinate of endpoint 1 + x-coordinate of endpoint 2)/2
= (2 + 0)/2
= 1
y-coordinate of midpoint = (y-coordinate of endpoint 1 + y-coordinate of endpoint 2)/2
= (4 + (-2))/2
= 1
So, the midpoint for the first segment is (1, 1).
Similarly, let's find the coordinates of the midpoint for the second segment:
x-coordinate of midpoint = (x-coordinate of endpoint 1 + x-coordinate of endpoint 2)/2
= (5 + 1)/2
= 3
y-coordinate of midpoint = (y-coordinate of endpoint 1 + y-coordinate of endpoint 2)/2
= (1 + 5)/2
= 3
So, the midpoint for the second segment is (3, 3).
Now, use the midpoint coordinates to find the slope of the line passing through them:
slope = (change in y-coordinates)/(change in x-coordinates)
= (3 - 1)/(3 - 1)
= 2/2
= 1
Therefore, the slope of the line containing the midpoints is 1.
(2) To find the value of k when the two given lines intersect at x = -8.4, substitute this value into the equations and solve for k.
For the first equation: -2x + y = k
Replacing x with -8.4, we have: -2(-8.4) + y = k
Simplifying: 16.8 + y = k
For the second equation: 0.5x + y = 14
Replacing x with -8.4, we have: 0.5(-8.4) + y = 14
Simplifying: -4.2 + y = 14
Now, we have two equations with y = k:
16.8 + y = k
-4.2 + y = 14
Simplifying the second equation further, we get: y = 14 + 4.2
= 18.2
Therefore, the value of k is 18.2.