To find the coordinates of the orthocenter of a triangle given the midpoints of its sides, we will follow these steps:
### Step 1: Understand the relationship between the midpoints and vertices
Let the midpoints of the sides of the triangle be denoted as:
- \( M_A = (0, 0) \)
- \( M_B = (1, 2) \)
- \( M_C = (-6, 3) \)
The vertices of the triangle can be represented as \( A, B, C \). The midpoints of the sides of the triangle are given by the following formulas:
- \( M_A = \frac{B + C}{2} \)
- \( M_B = \frac{A + C}{2} \)
- \( M_C = \frac{A + B}{2} \)
### Step 2: Set up equations for the vertices
From the midpoint formulas, we can derive the following equations:
1. \( (0, 0) = \frac{B + C}{2} \) → \( B + C = (0, 0) \)
2. \( (1, 2) = \frac{A + C}{2} \) → \( A + C = (2, 4) \)
3. \( (-6, 3) = \frac{A + B}{2} \) → \( A + B = (-12, 6) \)
### Step 3: Solve the equations
Now we have a system of equations:
1. \( B + C = (0, 0) \) → \( B = -C \)
2. \( A + C = (2, 4) \)
3. \( A + B = (-12, 6) \)
Substituting \( B = -C \) into the second and third equations:
- From \( A + C = (2, 4) \):
\[ A + C = (2, 4) \]
- From \( A + B = (-12, 6) \):
\[ A - C = (-12, 6) \]
Now we can solve these two equations:
1. \( A + C = (2, 4) \)
2. \( A - C = (-12, 6) \)
### Step 4: Add and subtract the equations
Adding the two equations:
\[ (A + C) + (A - C) = (2, 4) + (-12, 6) \]
\[ 2A = (-10, 10) \]
\[ A = (-5, 5) \]
Now substituting \( A \) back into one of the equations to find \( C \):
\[ -5 + C = 2 \Rightarrow C = 7 \]
\[ -5 + C = 4 \Rightarrow C = 9 \]
### Step 5: Find \( B \)
Using \( B = -C \):
\[ B = -(-5, 5) = (5, -5) \]
### Step 6: Identify the orthocenter
The orthocenter \( H \) of a triangle can be found using the vertices \( A, B, C \). The coordinates of the orthocenter can be calculated as:
\[ H = A + B + C \]
Substituting the values:
\[ H = (-5, 5) + (5, -5) + (7, 9) \]
\[ H = (-5 + 5 + 7, 5 - 5 + 9) \]
\[ H = (7, 9) \]
### Final Answer
The coordinates of the orthocenter of the triangle are \( (-5, 5) \).
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