IN triangle ABC, D(7,3) is the midpoint of AB, E(19,9) is the midpoint of BC, and F(5, 5) is the midpoint of AC, Find the coordinates of A, B, and C

To find the coordinates of points \( A \), \( B \), and \( C \) given that \( D(7,3) \), \( E(19,9) \), and \( F(5,5) \) are the midpoints of sides \( AB \), \( BC \), and \( AC \) of triangle \( ABC \), we can use the midpoint formula.

### Step 1: Set up the midpoint equations
Let's denote the coordinates of \( A \) as \( (x_1, y_1) \), \( B \) as \( (x_2, y_2) \), and \( C \) as \( (x_3, y_3) \).

The midpoint of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Using this formula for each midpoint:

1. **For \( D(7,3) \), the midpoint of \( AB \):**

   \[
   D = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = (7,3)
   \]

   This gives us the equations:
   \[
   \frac{x_1 + x_2}{2} = 7 \quad \text{and} \quad \frac{y_1 + y_2}{2} = 3
   \]
   Simplifying:
   \[
   x_1 + x_2 = 14 \quad \text{and} \quad y_1 + y_2 = 6
   \]

2. **For \( E(19,9) \), the midpoint of \( BC \):**

   \[
   E = \left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right) = (19,9)
   \]

   This gives us the equations:
   \[
   \frac{x_2 + x_3}{2} = 19 \quad \text{and} \quad \frac{y_2 + y_3}{2} = 9
   \]
   Simplifying:
   \[
   x_2 + x_3 = 38 \quad \text{and} \quad y_2 + y_3 = 18
   \]

3. **For \( F(5,5) \), the midpoint of \( AC \):**

   \[
   F = \left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right) = (5,5)
   \]

   This gives us the equations:
   \[
   \frac{x_1 + x_3}{2} = 5 \quad \text{and} \quad \frac{y_1 + y_3}{2} = 5
   \]
   Simplifying:
   \[
   x_1 + x_3 = 10 \quad \text{and} \quad y_1 + y_3 = 10
   \]

### Step 2: Solve the system of equations
Now we have three pairs of linear equations:

\[
\begin{align*}
(1) \quad & x_1 + x_2 = 14 \quad & (2) \quad & y_1 + y_2 = 6 \\
(3) \quad & x_2 + x_3 = 38 \quad & (4) \quad & y_2 + y_3 = 18 \\
(5) \quad & x_1 + x_3 = 10 \quad & (6) \quad & y_1 + y_3 = 10 \\
\end{align*}
\]

#### Solving for \( x_1 \), \( x_2 \), and \( x_3 \):

1. From equations (1) and (5):

\[
\begin{align*}
x_1 + x_2 &= 14 \quad & \text{(Equation 1)} \\
x_1 + x_3 &= 10 \quad & \text{(Equation 5)} \\
\end{align*}
\]
Subtract (5) from (1):
\[
x_2 - x_3 = 4 \quad \text{(Equation 7)}


In triangle ABC, D(7,3) is the midpoint of AB, E(19,9) is the midpoint of BC, and F(5,5) is the midpoint of AC. Find the coordinates of A, B, and C.

Learn how to determine the coordinates of vertices A, B, and C of a triangle given the midpoints of its sides. This problem involves key concepts in coordinate geometry, including the midpoint formula and solving systems of linear equations. Suitable for students in Grades 10-12.

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