In triangle DEF, A(2,3) is the midpoint of DE, B(4,1) is the midpoint of EF, and C(6,2) is the midpoint of DF. What are the coordinates of D, E, and F?
To find the coordinates of the points D, E, and F, we can use the midpoint formula.
The midpoint formula states that if we have two points (x₁, y₁) and (x₂, y₂) and the midpoint (x, y) is given, then the coordinates of the midpoint can be found using the following formulas:
x = (x₁ + x₂)/2
y = (y₁ + y₂)/2
Let's use these formulas to find the coordinates of each point.
1. First, we will find the coordinates of point D, which we don't have any information about. Let's assume the coordinates of D are (xD, yD).
2. Since A(2,3) is the midpoint of DE, we can use the midpoint formula to find the coordinates of point E:
x = (x₁ + x₂)/2
2 = (2 + xD)/2
4 = 2 + xD
xD = 2
y = (y₁ + y₂)/2
3 = (3 + yD)/2
6 = 3 + yD
yD = 3
So, the coordinates of point D are (2, 3).
3. We can proceed similarly to find the coordinates of point F. B(4,1) is the midpoint of EF:
x = (x₁ + x₂)/2
4 = (xD + xF)/2
8 = xD + xF
xF = 8 - xD
xF = 8 - 2
xF = 6
y = (y₁ + y₂)/2
1 = (yD + yF)/2
2 = yD + yF
yF = 2 - yD
yF = 2 - 3
yF = -1
So, the coordinates of point F are (6, -1).
Therefore, the coordinates of D, E, and F are:
D(2, 3)
E(2, 3)
F(6, -1)