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### Question 10:
"If \( T \) is the midpoint of \( SU \), find \( x \)."
- The two segments are \( 8x + 11 \) and \( 12x - 1 \).
Since \( T \) is the midpoint, the two segments are equal:
\[
8x + 11 = 12x - 1
\]
Solving for \( x \):
\[
8x - 12x = -1 - 11
\]
\[
-4x = -12
\]
\[
x = 3
\]

### Question 11:
"If \( G \) is the midpoint of \( FH \), find \( FG \)."
- The two segments are \( 11x - 7 \) and \( 3x + 9 \).
Since \( G \) is the midpoint, we have:
\[
11x - 7 = 3x + 9
\]
Solving for \( x \):
\[
11x - 3x = 9 + 7
\]
\[
8x = 16
\]
\[
x = 2
\]
Now, to find \( FG \), substitute \( x = 2 \) into \( FG = 11x - 7 \):
\[
FG = 11(2) - 7 = 22 - 7 = 15
\]

### Question 12:
"If \( R \) is the midpoint of \( QS \), find \( QS \)."
- The two segments are \( 5x - 3 \) and \( 21 - x \).
Since \( R \) is the midpoint, the two segments are equal:
\[
5x - 3 = 21 - x
\]
Solving for \( x \):
\[
5x + x = 21 + 3
\]
\[
6x = 24
\]
\[
x = 4
\]
Now, to find \( QS \), substitute \( x = 4 \) into both expressions and add them:
\[
QR = 5(4) - 3 = 20 - 3 = 17
\]
\[
RS = 21 - 4 = 17
\]
Thus, \( QS = 17 + 17 = 34 \).

### Question 13:
"If \( B \) is the midpoint of \( AC \), and \( AC = 8x - 20 \), find \( BC \)."
Since \( B \) is the midpoint, \( AB = BC \), so:
\[
AB = BC = \frac{AC}{2}
\]
Substitute \( AC = 8x - 20 \):
\[
BC = \frac{8x - 20}{2} = 4x - 10
\]

Now, the segment \( BC \) is also given as \( 3x - 1 \). Therefore, we can set the two equal:
\[
4x - 10 = 3x - 1
\]
Solving for \( x \):
\[
4x - 3x = -1 + 10
\]
\[
x = 9
\]
Now, substitute \( x = 9 \) into \( BC = 3x - 1 \):
\[
BC = 3(9) - 1 = 27 - 1 = 26
\]

Let me know if you'd like detailed explanations for any specific part! Here are some related questions to extend your learning:

1. What is the formula for the midpoint of a segment in coordinate geometry?
2. How do you calculate the length of a segment if given coordinates for both endpoints?
3. Can a midpoint problem be solved using a geometric approach instead of an algebraic one?
4. How do you interpret midpoint problems in three-dimensional geometry?
5. In real-life applications, where can the concept of a midpoint be used?

**Tip**: When solving midpoint problems, always remember that the two segments created by the midpoint are congruent, meaning they have equal lengths. This helps set up equations easily.

If T is the midpoint of SU, find x. If G is the midpoint of FH, find FG. If R is the midpoint of QS, find QS. If B is the midpoint of AC, and AC = 8x - 20, find BC.

This page covers algebraic solutions for midpoint problems involving linear equations. Learn how to solve for x when given midpoint conditions, including examples like 'If T is the midpoint of SU, find x' and 'If G is the midpoint of FH, find FG'. Suitable for Grades 8-10.

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