The coordinates of the vertices of a parallelogram are given. Find the possible coordinates for the fourth vertex.
L(0,4), M(6,0),N(2,4)
To find the possible coordinates for the fourth vertex of the parallelogram, we can use the properties of a parallelogram.
One property of a parallelogram is that opposite sides are parallel, which means that the slope of one side is equal to the slope of the opposite side.
Let's use points L(0,4) and M(6,0) to find the slope of one side of the parallelogram.
Slope of LM = (change in y) / (change in x)
= (0 - 4) / (6 - 0)
= -4 / 6
= -2/3
The opposite side of the parallelogram will have the same slope.
Now, let's use the slope and the coordinates of point N(2,4) to find the equation of the line passing through N.
Equation of a line in slope-intercept form: y = mx + b
Since we know the slope (-2/3) and the coordinates of N(2,4), we can substitute those values into the equation:
4 = (-2/3)(2) + b
Next, we solve for b:
4 = -4/3 + b
4 + 4/3 = b
12/3 + 4/3 = b
16/3 = b
Therefore, the equation of the line passing through N is y = (-2/3)x + 16/3.
Now, we can use this equation to find the possible coordinates for the fourth vertex. Let's call this point P(x, y).
Substituting the equation of the line into the equation of y, we get:
(-2/3)x + 16/3 = y
Now, we know that point P is also on the line LM. So we can use the equation of slope to find the relationship between x and y.
Slope of LM = -2/3
Slope of PM = (change in y) / (change in x)
Since points L and M are on the line LM, we can substitute their coordinates into the equation of slope:
(-2/3) = (y - 4) / (x - 0)
(-2/3) = (y - 4) / x
Cross-multiplying, we get:
-2x = 3(y - 4)
-2x = 3y - 12
3y = -2x + 12
y = (-2/3)x + 4
Now, we have a system of equations:
1) y = (-2/3)x + 16/3
2) y = (-2/3)x + 4
We can solve this system of equations to find the coordinates of point P, which represents the possible coordinates for the fourth vertex.
Solving the system, we have:
(-2/3)x + 16/3 = (-2/3)x + 4
We can see that both equations have the same slope (-2/3). This means that the lines are parallel and will never intersect. Therefore, there are no possible coordinates for the fourth vertex, and the given points L(0,4), M(6,0), and N(2,4) do not form a parallelogram.