Question

Line a passes through points (1, 18) and (10, 10). Line b passes through points (9, 10) and (1, 1). Are line a and line b parallel or perpendicular?

Answer 1

To determine if the lines are parallel or perpendicular, we will first find the slopes of both lines.

**Slope of Line A:**
Line A passes through points (1, 18) and (10, 10). The slope \( m_A \) is calculated using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the coordinates of the points:

\[
m_A = \frac{10 - 18}{10 - 1} = \frac{-8}{9}
\]

**Slope of Line B:**
Line B passes through points (9, 10) and (1, 1). The slope \( m_B \) is calculated similarly:

\[
m_B = \frac{1 - 10}{1 - 9} = \frac{-9}{-8} = \frac{9}{8}
\]

**Determine Relationship between the Slopes:**
Two lines are parallel if their slopes are equal and perpendicular if the product of their slopes is -1.

To check if they are perpendicular:

\[
m_A \cdot m_B = \left(-\frac{8}{9}\right) \cdot \left(\frac{9}{8}\right)
\]

Calculating the product:

\[
m_A \cdot m_B = -\frac{8 \cdot 9}{9 \cdot 8} = -1
\]

Since the product of the slopes is -1, this means that Line A and Line B are **perpendicular**.