Question

name: yoaca
date: 9/10/25
per: 2
unit 1: geometry basics
homework 2: segment additio
this is a 2-page document!
use the diagram below to
answer questions 1 and 2.
diagram: l---m---n

1. if lm = 22 and mn = 15, find l...
2. if ln = 54 and lm = 31, find mn.

left side: \=36, find the value of x.\ with diagram s---t, 6x+1 and x+7

1. if df = 9x - 39, find ef.

diagram: d---47---e---3x+10---f

1. if hj = 7x - 27, find the value of x.

diagram: h---3x-5---i---x-1---j
bottom: x = , bc = , cd = __

Explanation:

Problem 6:

Step1: Apply Segment Addition Postulate

From the diagram, \( HJ = HI + IJ \). We know \( HI = 3x - 5 \), \( IJ = x - 1 \), and \( HJ = 7x - 27 \). So we set up the equation:
\( 7x - 27=(3x - 5)+(x - 1) \)

Step2: Simplify the right - hand side

Simplify \( (3x - 5)+(x - 1) \):
\( 3x - 5+x - 1 = 4x-6 \)
So the equation becomes \( 7x - 27 = 4x-6 \)

Step3: Solve for \( x \)

Subtract \( 4x \) from both sides:
\( 7x-4x - 27=4x - 4x-6 \)
\( 3x - 27=-6 \)
Add 27 to both sides:
\( 3x-27 + 27=-6 + 27 \)
\( 3x=21 \)
Divide both sides by 3:
\( x = \frac{21}{3}=7 \)

Step1: Apply Segment Addition Postulate

From the diagram, \( DF=DE + EF \). We know \( DE = 47 \), \( EF = 3x + 10 \), and \( DF=9x - 39 \). So we set up the equation:
\( 9x - 39=47+(3x + 10) \)

Step2: Simplify the right - hand side

Simplify \( 47+(3x + 10) \):
\( 3x+47 + 10=3x + 57 \)
So the equation becomes \( 9x - 39=3x + 57 \)

Step3: Solve for \( x \)

Subtract \( 3x \) from both sides:
\( 9x-3x - 39=3x-3x + 57 \)
\( 6x - 39=57 \)
Add 39 to both sides:
\( 6x-39 + 39=57 + 39 \)
\( 6x=96 \)
Divide both sides by 6:
\( x=\frac{96}{6}=16 \)

Step4: Find \( EF \)

Substitute \( x = 16 \) into the expression for \( EF \) (\( EF = 3x + 10 \)):
\( EF=3\times16 + 10=48 + 10 = 58 \)

Step1: Apply Segment Addition Postulate

From the diagram (for points \( L\), \( M\), \( N\)), \( LN=LM + MN \). We know \( LN = 54 \) and \( LM = 31 \). We want to find \( MN \).
Rearrange the formula to solve for \( MN \): \( MN=LN - LM \)

Step2: Substitute the values

Substitute \( LN = 54 \) and \( LM = 31 \) into the formula:
\( MN = 54-31=23 \) (Note: The original hand - written answer of 85 is incorrect. The correct calculation is \( 54 - 31=23\))

Answer:

\( x = 7 \)

Problem 4: