for the line segment DE one edpoint is D(3,1) and the midpoint is M (0,-4). find the coordinates of endpoint E.
To find the coordinates of endpoint E of line segment DE, we can use the midpoint formula.
The midpoint formula states that the coordinates of the midpoint (M) of a line segment with endpoints (x₁, y₁) and (x₂, y₂) can be found using the following formulas:
x-coordinate of M = (x₁ + x₂) / 2
y-coordinate of M = (y₁ + y₂) / 2
We already know the coordinates of endpoint D, which are D(3, 1), and the coordinates of the midpoint, which are M(0, -4).
Substituting these values into the midpoint formula, we can solve for the coordinates of endpoint E.
x-coordinate of M = (x₁ + x₂) / 2
0 = (3 + x₂) / 2
0 = 3 + x₂
Solving for x₂:
x₂ = -3
y-coordinate of M = (y₁ + y₂) / 2
-4 = (1 + y₂) / 2
-8 = 1 + y₂
Solving for y₂:
y₂ = -9
Therefore, the coordinates of endpoint E are E(-3, -9).
To find the coordinates of endpoint E, we need to use the midpoint formula. The midpoint formula states that the coordinates of the midpoint (M) between two endpoints (D and E) on a line segment can be found by taking the average of the x-coordinates and the average of the y-coordinates.
Let's denote the coordinates of endpoint E as (x, y).
We are given that the coordinates of endpoint D are D(3,1) and the coordinates of the midpoint M are M(0,-4).
Using the midpoint formula, we can set up the following equations:
x-coordinate equation:
(x-coordinate of D + x-coordinate of E) / 2 = x-coordinate of M
(3 + x) / 2 = 0
Solving this equation will give us the x-coordinate of endpoint E.
y-coordinate equation:
(y-coordinate of D + y-coordinate of E) / 2 = y-coordinate of M
(1 + y) / 2 = -4
Solving this equation will give us the y-coordinate of endpoint E.
Now, let's solve these equations:
x-coordinate equation:
(3 + x) / 2 = 0
Multiply both sides of the equation by 2:
3 + x = 0
Subtracting 3 from both sides:
x = -3
Therefore, the x-coordinate of endpoint E is -3.
y-coordinate equation:
(1 + y) / 2 = -4
Multiply both sides of the equation by 2:
1 + y = -8
Subtracting 1 from both sides:
y = -9
Therefore, the y-coordinate of endpoint E is -9.
Therefore, the coordinates of endpoint E are (-3, -9).
Since M is halfway from D to E, figure the difference between M and D and go that far again past M:
M-D = (-3,-5)
M+(-3,-5) = (-3,-9) = E