9. if $\overline{lk} \cong \overline{mk}$, $lk = 7x - 10$, $kn = x + 3$, $mn = 9x - 11$, and $kj = 28$, find $lj$.
10. if $t$ is the midpoint of $\overline{su}$, find $x$. segments: $st = 8x + 11$, $tu = 12x - 1$
11. if $g$ is the midpoint of $\overline{fh}$, find $fg$. segments: $fg = 11x - 7$, $gh = 3x + 9$
12. if $r$ is the midpoint of $\overline{qs}$, find $qs$. segments: $qr = 5x - 3$, $rs = 21 - x$
13. if $b$ is the midpoint of $\overline{ac}$, and $ac = 8x - 20$, find $bc$. segment: $ab = 3x - 1$
14. if $\overline{ef}$ bisects $\overline{cd}$, $cg = 5x - 1$, $gd = 7x - 13$, $ef = 6x - 4$, and $gf = 13$, find $eg$.
15. if $r$ is the midpoint of $\overline{qs}$, $rs = 2x - 4$, $st = 4x - 1$, and $rt = 8x - 43$, find $qs$.

Question

Accepted Answer

### Problem 10:
# Explanation:
## Step1: Use midpoint definition
Since $ T $ is the midpoint of $ \overline{SU} $, $ ST = TU $. So $ 8x + 11 = 12x - 1 $.
## Step2: Solve for $ x $
Subtract $ 8x $ from both sides: $ 11 = 4x - 1 $.
Add 1 to both sides: $ 12 = 4x $.
Divide by 4: $ x = 3 $.
# Answer: $ x = 3 $

### Problem 11:
# Explanation:
## Step1: Use midpoint definition
Since $ G $ is the midpoint of $ \overline{FH} $, $ FG = GH $. So $ 11x - 7 = 3x + 9 $.
## Step2: Solve for $ x $
Subtract $ 3x $ from both sides: $ 8x - 7 = 9 $.
Add 7 to both sides: $ 8x = 16 $.
Divide by 8: $ x = 2 $.
## Step3: Find $ FG $
Substitute $ x = 2 $ into $ FG = 11x - 7 $: $ FG = 11(2) - 7 = 22 - 7 = 15 $.
# Answer: $ FG = 15 $

### Problem 12:
# Explanation:
## Step1: Use midpoint definition
Since $ R $ is the midpoint of $ \overline{QS} $, $ QR = RS $. So $ 5x - 3 = 21 - x $.
## Step2: Solve for $ x $
Add $ x $ to both sides: $ 6x - 3 = 21 $.
Add 3 to both sides: $ 6x = 24 $.
Divide by 6: $ x = 4 $.
## Step3: Find $ QS $
$ QS = QR + RS = (5x - 3) + (21 - x) = 4x + 18 $. Substitute $ x = 4 $: $ 4(4) + 18 = 16 + 18 = 34 $.
# Answer: $ QS = 34 $

### Problem 13:
# Explanation:
## Step1: Use midpoint definition
Since $ B $ is the midpoint of $ \overline{AC} $, $ AB = BC $ and $ AC = 2AB $. Given $ AB = 3x - 1 $ and $ AC = 8x - 20 $, so $ 8x - 20 = 2(3x - 1) $.
## Step2: Solve for $ x $
Expand: $ 8x - 20 = 6x - 2 $.
Subtract $ 6x $: $ 2x - 20 = -2 $.
Add 20: $ 2x = 18 $.
Divide by 2: $ x = 9 $.
## Step3: Find $ BC $
$ BC = AB = 3x - 1 $. Substitute $ x = 9 $: $ 3(9) - 1 = 27 - 1 = 26 $.
# Answer: $ BC = 26 $

### Problem 14:
# Explanation:
## Step1: Use bisector definition
Since $ \overline{EF} $ bisects $ \overline{CD} $, $ CG = GD $. So $ 5x - 1 = 7x - 13 $.
## Step2: Solve for $ x $
Subtract $ 5x $: $ -1 = 2x - 13 $.
Add 13: $ 12 = 2x $.
Divide by 2: $ x = 6 $.
## Step3: Find $ EF $
Substitute $ x = 6 $ into $ EF = 6x - 4 $: $ EF = 6(6) - 4 = 36 - 4 = 32 $.
## Step4: Find $ EG $
Since $ EF = EG + GF $ and $ GF = 13 $, $ EG = EF - GF = 32 - 13 = 19 $.
# Answer: $ EG = 19 $

### Problem 15:
# Explanation:
## Step1: Use midpoint definition
Since $ R $ is the midpoint of $ \overline{QS} $, $ QR = RS = 2x - 4 $. Also, $ RT = RS + ST $, so $ 8x - 43 = (2x - 4) + (4x - 1) $.
## Step2: Solve for $ x $
Simplify right side: $ 8x - 43 = 6x - 5 $.
Subtract $ 6x $: $ 2x - 43 = -5 $.
Add 43: $ 2x = 38 $.
Divide by 2: $ x = 19 $.
## Step3: Find $ QS $
$ QS = 2 \times RS = 2(2x - 4) $. Substitute $ x = 19 $: $ 2(2(19) - 4) = 2(38 - 4) = 2(34) = 68 $.
# Answer: $ QS = 68 $

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QUESTION IMAGE

Question

Explanation:

Problem 10:

Step1: Use midpoint definition

Step2: Solve for \( x \)

Step1: Use midpoint definition

Step2: Solve for \( x \)

Step3: Find \( FG \)

Step1: Use midpoint definition

Step2: Solve for \( x \)

Step3: Find \( QS \)

Answer:

Problem 11: