  Find the missing endpoint if S is the midpoint RT. 10.  R(-9, 4) and S(2, -1);  Find T. 11.  S(-4, -6) and T(-7, -3);  Find R. 12.  B is the midpoint of ACand E is the midpoint of BD.  If A(-9, -4), C(-1, 6), and E(-4, -3), find  the coordinates of D. 

Let's solve each problem step by step.

### Problem 10: Finding the missing endpoint \( T \) if \( S \) is the midpoint of \( RT \).

Given:
- \( R(-9, 4) \)
- \( S(2, -1) \)

Let the coordinates of \( T \) be \( (x, y) \).

Since \( S \) is the midpoint of \( RT \), the coordinates of \( S \) are the average of the coordinates of \( R \) and \( T \):
\[
S = \left( \frac{x_R + x_T}{2}, \frac{y_R + y_T}{2} \right)
\]
Substitute the known values:
\[
2 = \frac{-9 + x}{2}
\]
\[
-1 = \frac{4 + y}{2}
\]

Solving for \( x \) and \( y \):

1. \( 2 = \frac{-9 + x}{2} \)  
   Multiply both sides by 2:
   \[
   4 = -9 + x
   \]
   Add 9 to both sides:
   \[
   x = 13
   \]

2. \( -1 = \frac{4 + y}{2} \)  
   Multiply both sides by 2:
   \[
   -2 = 4 + y
   \]
   Subtract 4 from both sides:
   \[
   y = -6
   \]

So, the coordinates of \( T \) are \( (13, -6) \).

### Problem 11: Finding the missing endpoint \( R \) if \( S \) is the midpoint of \( RT \).

Given:
- \( S(-4, -6) \)
- \( T(-7, -3) \)

Let the coordinates of \( R \) be \( (x, y) \).

Since \( S \) is the midpoint of \( RT \), we have:
\[
S = \left( \frac{x + x_T}{2}, \frac{y + y_T}{2} \right)
\]
Substitute the known values:
\[
-4 = \frac{x + (-7)}{2}
\]
\[
-6 = \frac{y + (-3)}{2}
\]

Solving for \( x \) and \( y \):

1. \( -4 = \frac{x - 7}{2} \)  
   Multiply both sides by 2:
   \[
   -8 = x - 7
   \]
   Add 7 to both sides:
   \[
   x = -1
   \]

2. \( -6 = \frac{y - 3}{2} \)  
   Multiply both sides by 2:
   \[
   -12 = y - 3
   \]
   Add 3 to both sides:
   \[
   y = -9
   \]

So, the coordinates of \( R \) are \( (-1, -9) \).

### Problem 12: Finding the coordinates of \( D \) given that \( B \) is the midpoint of \( AC \) and \( E \) is the midpoint of \( BD \).

Given:
- \( A(-9, -4) \)
- \( C(-1, 6) \)
- \( E(-4, -3) \)

Step 1: Find \( B \), the midpoint of \( AC \).

\[
B = \left( \frac{-9 + (-1)}{2}, \frac{-4 + 6}{2} \right)
\]
Simplify:
\[
B = \left( \frac{-10}{2}, \frac{2}{2} \right) = (-5, 1)
\]

Step 2: Since \( E \) is the midpoint of \( BD \), let \( D(x, y) \). We have:
\[
E = \left( \frac{-5 + x}{2}, \frac{1 + y}{2} \right)
\]
Substitute the coordinates of \( E(-4, -3) \):
\[
-4 = \frac{-5 + x}{2}
\]
\[
-3 = \frac{1 + y}{2}
\]

Solve for \( x \) and \( y \):

1. \( -4 = \frac{-5 + x}{2} \)  
   Multiply both sides by 2:
   \[
   -8 = -5 + x
   \]
   Add 5 to both sides:
   \[
   x = -3
   \]

2. \( -3 = \frac{1 + y}{2} \)  
   Multiply both sides by 2:
   \[
   -6 = 1 + y
   \]
   Subtract 1 from both sides:
   \[
   y = -7
   \]

So, the coordinates of \( D \) are \( (-3, -7) \).

---

### Summary of Answers:

1. **Problem 10**: \( T(13, -6) \)
2. **Problem 11**: \( R(-1, -9) \)
3. **Problem 12**: \( D(-3, -7) \)

Would you like a detailed explanation of any step? Here are some related questions:

1. How do you find the midpoint of a line segment?
2. How do you use the midpoint formula to find a missing endpoint?
3. What are some common mistakes when applying the midpoint formula?
4. How does the midpoint formula relate to coordinate geometry?
5. Can you solve problems involving midpoints and distance simultaneously?

**Tip:** Always double-check your midpoint and endpoint calculations by substituting the results back into the midpoint formula to ensure accuracy.

Learn how to find missing endpoints in coordinate geometry problems using the midpoint formula. Detailed solutions for problems involving midpoint calculations with coordinates of points R, S, T, A, C, B, and E. Suitable for Grades 9-12 students.

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