Question

b = ______ 5. show all work.

points x, y, and z are collinear, and y is
the midpoint of \\(\overline{xz}\\). find the value of b.
(lesson 3-5)

image of a line segment with x at one end, y in the middle, z at the other end. the length from x to y is \\(2b + 7\\), and from y to z is \\(3b - 4\\)

1. state the rules you used to answer this question.

three lines have these equations:
line m: \\(y = \frac{2}{3}x - 7\\)
line n: \\(y = -\frac{2}{3}(x + 1)\\)
line p: \\(y = -\frac{3}{2}x + 4\\)

identify the relationship between each pair
of lines. (lesson 3-8)

lines	m and n	m and p	n and p
perpendicular
neither

blank lines for writing the rules

Explanation:

Problem 5 (Finding \( b \) when \( Y \) is the midpoint of \( \overline{XZ} \))

Step 1: Recall Midpoint Definition

Since \( Y \) is the midpoint of \( \overline{XZ} \), the length of \( \overline{XY} \) equals the length of \( \overline{YZ} \). So, \( 2b + 7 = 3b - 4 \).

Step 2: Solve for \( b \)

Subtract \( 2b \) from both sides: \( 7 = b - 4 \).
Add 4 to both sides: \( b = 7 + 4 = 11 \).

Answer:

\( b = 11 \)

Problem 6 (Relationships Between Lines)

First, recall the rules for parallel and perpendicular lines in slope - intercept form (\( y = mx + c \), where \( m \) is the slope):

• Parallel Lines: Two lines are parallel if their slopes are equal (\( m_1=m_2 \)).
• Perpendicular Lines: Two lines are perpendicular if the product of their slopes is \( - 1 \) (\( m_1\times m_2=-1 \)). If neither of these conditions is met, the lines are neither parallel nor perpendicular.

Step 1: Identify the slopes of each line

• For Line \( m \): \( y=\frac{2}{3}x - 7 \), the slope \( m_m=\frac{2}{3} \).
• For Line \( n \): \( y =-\frac{2}{3}(x + 1)=-\frac{2}{3}x-\frac{2}{3} \), the slope \( m_n=-\frac{2}{3} \).
• For Line \( p \): \( y =-\frac{3}{2}x + 4 \), the slope \( m_p=-\frac{3}{2} \).

Step 2: Analyze each pair of lines

• Lines \( m \) and \( n \):
• \( m_m=\frac{2}{3} \) and \( m_n =-\frac{2}{3} \). Since \( \frac{2}{3}

eq-\frac{2}{3} \), they are not parallel.

• The product of their slopes: \( \frac{2}{3}\times(-\frac{2}{3})=-\frac{4}{9}

eq - 1 \). So, they are neither parallel nor perpendicular.

• Lines \( m \) and \( p \):
• \( m_m=\frac{2}{3} \) and \( m_p=-\frac{3}{2} \). The product of their slopes: \( \frac{2}{3}\times(-\frac{3}{2})=-1 \). So, they are perpendicular.
• Lines \( n \) and \( p \):
• \( m_n=-\frac{2}{3} \) and \( m_p=-\frac{3}{2} \). Since \( -\frac{2}{3}

eq-\frac{3}{2} \), they are not parallel.

• The product of their slopes: \( (-\frac{2}{3})\times(-\frac{3}{2}) = 1

eq - 1 \). So, they are neither parallel nor perpendicular.

Now, fill in the table:

Lines	parallel	perpendicular	neither
\( m \) and \( p \)		✔️
\( n \) and \( p \)			✔️

Rules Used:

• For two lines in the form \( y = m_1x + c_1 \) and \( y=m_2x + c_2 \):
• Parallel: \( m_1=m_2 \)
• Perpendicular: \( m_1\times m_2=-1 \)
• If neither condition is met, the lines are neither parallel nor perpendicular.