With the mid segment of the triangle MNP as M (-1, 3), N (4, 0), P (-2, -2) the coordinates of the triangle are
- (-7, 1), (5, 5), and (3, -5)
How to find the coordinates of the triangle when the mid segment is known
The coordinates is calculated from the midsegment using the formula
X= (x₂ + x₁)/2
Y = (y₂ + y₁)/2
Vertex 1 and 2: point M (-1, 3),
X
-1 = (x₂ + x₁)/2
-2 = x₂ + x₁
Y
3 = (y₂ + y₁)/2
6 = y₂ + y₁
Vertex 2 and 3: point N (4, 0)
X
4 = (x₂ + x₃)/2
8 = x₂ + x₃
Y
0 = (y₂ + y₃)/2
0 = y₂ + y₃
Vertex 1 and 3: point P(-2, -2)
X
-2 = (x₃ + x₁)/2
-4 = x₃ + x₁
Y
-2 = (y₃ + y₁)/2
-4 = y₃ + y₁
The equations
X
-2 = x₂ + x₁
8 = x₂ + x₃
-4 = x₃ + x₁
Y
6 = y₂ + y₁
0 = y₂ + y₃
-4 = y₃ + y₁
from -2 = x₂ + x₁, we find x₁ = -2 - x₂
plugging in x₁ = -2 - x₂ to -4 = x₃ + x₁ gives
-4 = x₃ + -2 - x₂
-2 = x₃ - x₂
then we have
8 = x₂ + x₃
-2 = x₃ - x₂
solving simultaneously
x₃ = 3, x₂ = 5 then x₁ = -2 - x₂ = -7
from 0 = y₂ + y₃, we find that y₂ = -y₃
plugging in y₂ = -y₃ into 6 = y₂ + y₁ gives
6 = -y₃ + y₁
then we have
-4 = y₃ + y₁
6 = -y₃ + y₁
solving simultaneously gives
y₁ = 1, y₃ = -5, then y₂ = -y₃ = -(-4) = 5
The vertex of the triangle are
(-7, 1), (5, 5), and (3, -5)
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