geometry
part 2: segment addition (honors)
solve for x.
1) u 3 t x - 1 s 2 r
←-------- 2x - 7 --------→
2) g x + 17 f 10 e x + 18 d
←------------ 31 ------------→
3) u 10 t x - 1 s x - 2 r
←------------ 13 ------------→
4) i 2x + 13 h 10 g 2 f
←------------ 20 + x ------------→
5) w x + 5 v u 7 t
←-------- 6 --------→
←-------- x + 10 --------→ (from u to t)
6) f 2x + 23 g h 6 i
←-------- x + 24 --------→ (from g to i)
←------------ 14 ------------→ (from f to h)
name mariell id:
date 8 sep 25 period 2

Question

geometry
part 2: segment addition (honors)
solve for x.
1) u  3  t  x - 1  s  2  r
    ←-------- 2x - 7 --------→
2) g  x + 17  f  10  e  x + 18  d
    ←------------ 31 ------------→
3) u  10  t  x - 1  s  x - 2  r
    ←------------ 13 ------------→
4) i  2x + 13  h  10  g  2  f
    ←------------ 20 + x ------------→
5) w  x + 5  v  u  7  t
    ←-------- 6 --------→
    ←-------- x + 10 --------→ (from u to t)
6) f  2x + 23  g  h  6  i
    ←-------- x + 24 --------→ (from g to i)
    ←------------ 14 ------------→ (from f to h)
name mariell id:
date 8 sep 25 period 2

Accepted Answer

### Problem 1:
# Explanation:
## Step1: Apply Segment Addition Postulate
The total length $ UR = UT + TS + SR $, so $ 2x - 7 = 3 + (x - 1) + 2 $
## Step2: Simplify the equation
Simplify right side: $ 3 + x - 1 + 2 = x + 4 $. So $ 2x - 7 = x + 4 $
## Step3: Solve for $ x $
Subtract $ x $ from both sides: $ x - 7 = 4 $. Add 7 to both sides: $ x = 11 $
# Answer: $ x = 11 $

### Problem 2:
# Explanation:
## Step1: Apply Segment Addition Postulate
$ GD = GF + FE + ED $, so $ 31 = (x + 17) + 10 + (x + 18) $
## Step2: Simplify the equation
Simplify right side: $ x + 17 + 10 + x + 18 = 2x + 45 $. So $ 31 = 2x + 45 $
## Step3: Solve for $ x $
Subtract 45: $ -14 = 2x $. Divide by 2: $ x = -7 $
# Answer: $ x = -7 $

### Problem 3:
# Explanation:
## Step1: Apply Segment Addition Postulate
$ UR = UT + TS + SR $, so $ 13 = 10 + (x - 1) + (x - 2) $
## Step2: Simplify the equation
Simplify right side: $ 10 + x - 1 + x - 2 = 2x + 7 $. So $ 13 = 2x + 7 $
## Step3: Solve for $ x $
Subtract 7: $ 6 = 2x $. Divide by 2: $ x = 3 $
# Answer: $ x = 3 $

### Problem 4:
# Explanation:
## Step1: Apply Segment Addition Postulate
$ IF = IH + HG + GF $, so $ 20 + x = (2x + 13) + 10 + 2 $
## Step2: Simplify the equation
Simplify right side: $ 2x + 13 + 10 + 2 = 2x + 25 $. So $ 20 + x = 2x + 25 $
## Step3: Solve for $ x $
Subtract $ x $: $ 20 = x + 25 $. Subtract 25: $ x = -5 $
# Answer: $ x = -5 $

### Problem 5:
# Explanation:
## Step1: Analyze the segments
From $ W $ to $ U $ is 6, from $ U $ to $ T $ is 7, so from $ W $ to $ T $ is $ 6 + 7 = 13 $. Also, from $ U $ to $ T $ can be expressed as $ (x + 10) - (x + 5) $? Wait, better: From $ V $ to $ T $ is $ x + 10 $, and $ VU = 6 - (x + 5) $? Wait, correct approach: $ WU = 6 $, $ UT = 7 $, so $ WT = 13 $. Also, $ VU = 6 - (x + 5) $, but maybe better: $ x + 10 = (x + 5) + 7 $? Wait, no. Wait, the segment from $ U $ to $ T $ is 7, and from $ U $ to $ T $ is also $ (x + 10) - (x + 5) $? Wait, no, let's re - express. The length from $ V $ to $ U $ is $ 6 - (x + 5) $, but actually, the segment from $ W $ to $ V $ is $ x + 5 $, from $ V $ to $ U $ is some length, from $ U $ to $ T $ is 7. And from $ U $ to $ T $ is also $ (x + 10) - (x + 5) $? Wait, no, the correct equation: $ x + 10 = (x + 5) + 7 $? Wait, that would be $ x + 10 = x + 12 $, which is wrong. Wait, maybe the segment from $ W $ to $ U $ is 6, and from $ U $ to $ T $ is $ x + 10 - (x + 5) $? Wait, no, let's look at the diagram again. The segment $ WU = 6 $, $ UT = 7 $, so $ WT = 13 $. Also, $ VU $ is part of $ WU $, and $ UT = x + 10 $? Wait, no, the problem says: $ W $ to $ V $ is $ x + 5 $, $ V $ to $ U $ is some point, $ U $ to $ T $ is 7. And the length from $ U $ to $ T $ is also $ x + 10 - (x + 5) $? Wait, I think I made a mistake. Let's start over. The segment $ WU = 6 $, and $ UT = x + 10 $, but also $ UT = 7 + (WU - (x + 5)) $? No, better: The length from $ V $ to $ T $ is $ x + 10 $, and $ VT = VU + UT $. $ VU = WU - WV = 6 - (x + 5) $, and $ UT = 7 $. So $ x + 10=(6-(x + 5))+7 $
## Step2: Simplify the equation
Simplify right side: $ 6 - x - 5 + 7=8 - x $. So $ x + 10 = 8 - x $
## Step3: Solve for $ x $
Add $ x $ to both sides: $ 2x+10 = 8 $. Subtract 10: $ 2x=-2 $. Divide by 2: $ x=-1 $
# Answer: $ x = - 1 $

### Problem 6:
# Explanation:
## Step1: Apply Segment Addition Postulate
We know that $ FI = 14 $, and $ GI=x + 24 $, $ HI = 6 $, $ FG=2x + 23 $. Also, $ GI=FG + GH $, but $ HI = 6 $, and $ FI = 14 $, so $ FG=14 - 6=8 $? Wait, no. Wait, the segment $ FI = 14 $, $ HI = 6 $, so $ FH = 14 - 6 = 8 $. And $ FH=FG + GH $, but $ FG = 2x+23 $, and $ GI=x + 24 $, $ HI = 6 $, so $ FG=GI - HI=(x + 24)-6=x + 18 $. So $ 2x + 23=x + 18 $
## Step2: Solve for $ x $
Subtract $ x $ from both sides: $ x+23 = 18 $. Subtract 23: $ x=-5 $
Wait, let's check again. The segment $ FI = 14 $, $ HI = 6 $, so $ FH=14 - 6 = 8 $. And $ FH = FG $, because $ FG $ is from $ F $ to $ G $, and $ FH $ is from $ F $ to $ H $? No, the diagram: $ F---G---H---I $, with $ FG = 2x + 23 $, $ GH $ is some length, $ HI = 6 $, $ GI=x + 24 $, and $ FI = 14 $. So $ GI=GH + HI $, and $ FI=FG + GH + HI $. So $ 14=(2x + 23)+GH + 6 $, and $ x + 24=GH + 6 $, so $ GH=x + 18 $. Substitute into first equation: $ 14=2x + 23+x + 18+6 $? No, that's wrong. Wait, correct: $ FI = FG + GI - HI $? No, better: $ FI = FG + (GI - HI) $? No, let's use the segment addition: $ FI=FG + GH + HI $, and $ GI=GH + HI $, so $ FI=FG + GI $. Wait, $ FI = 14 $, $ GI=x + 24 $, $ FG=2x + 23 $, so $ 14=(2x + 23)+(x + 24)-6 $? No, I think the correct equation is $ x + 24=(2x + 23)+6 $? No, that would be $ x + 24=2x + 29 $, $ -x = 5 $, $ x=-5 $. Let's verify: If $ x=-5 $, $ FG=2*(-5)+23 = 13 $, $ GI=-5 + 24 = 19 $, $ HI = 6 $, then $ FG+HI=13 + 6 = 19=GI $, and $ FI=FG + GH $, but $ GH=GI - HI=19 - 6 = 13 $, and $ FI=FG + GH=13 + 13 = 26 $? No, that's not 14. Wait, I made a mistake. The correct approach: $ FI = 14 $, $ HI = 6 $, so $ FH=14 - 6 = 8 $. And $ FH = FG $, so $ 2x + 23=8 $, $ 2x=-15 $, $ x=-7.5 $. Wait, no, the diagram says $ FI = 14 $, and $ GI=x + 24 $, $ HI = 6 $, so $ FG=FI - GI + HI $? No, I think the original problem's diagram: $ F---G---H---I $, with $ FG = 2x + 23 $, $ GH $ is between $ G $ and $ H $, $ HI = 6 $, $ GI=x + 24 $ (from $ G $ to $ I $), and $ FI = 14 $ (from $ F $ to $ I $). So $ FI=FG + GI - HI $? No, $ FI=FG + GH $, and $ GI=GH + HI $, so $ GH=GI - HI=x + 24 - 6=x + 18 $. Then $ FI=FG + GH=(2x + 23)+(x + 18)=3x + 41 $. But $ FI = 14 $, so $ 3x+41 = 14 $, $ 3x=-27 $, $ x=-9 $. Wait, this is confusing. Let's start over with the correct segment addition. The segment from $ F $ to $ I $ is 14. The segment from $ G $ to $ I $ is $ x + 24 $, and the segment from $ F $ to $ G $ is $ 2x + 23 $, and the segment from $ H $ to $ I $ is 6. So $ FG + (GI - HI)=FI $, so $ (2x + 23)+(x + 24 - 6)=14 $, $ 2x + 23+x + 18 = 14 $, $ 3x+41 = 14 $, $ 3x=-27 $, $ x=-9 $. Let's check: $ FG=2*(-9)+23 = 5 $, $ GI=-9 + 24 = 15 $, $ HI = 6 $, so $ GH=GI - HI=15 - 6 = 9 $, $ FI=FG + GH=5 + 9 = 14 $. Yes, that works.
## Step1: Set up the equation
$ FI = FG + (GI - HI) $, so $ 14=(2x + 23)+(x + 24 - 6) $
## Step2: Simplify the equation
Simplify right side: $ 2x + 23+x + 18=3x + 41 $. So $ 14 = 3x + 41 $
## Step3: Solve for $ x $
Subtract 41: $ -27 = 3x $. Divide by 3: $ x=-9 $
# Answer: $ x = - 9 $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1:

Step1: Apply Segment Addition Postulate

Step2: Simplify the equation

Step3: Solve for \( x \)

Step1: Apply Segment Addition Postulate

Step2: Simplify the equation

Step3: Solve for \( x \)

Step1: Apply Segment Addition Postulate

Step2: Simplify the equation

Step3: Solve for \( x \)

Answer:

Problem 2: