Question

Line t passes through points (97, 29) and (39, -19). Line u passes through points (23, 10) and (98, 2). Are line t and line u parallel or perpendicular?

Answer 1

To determine whether line \( t \) and line \( u \) are parallel or perpendicular, we first need to calculate the slopes of both lines.

**Slope of Line \( t \)**:

The slope \( m \) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For line \( t \), the points are \( (97, 29) \) and \( (39, -19) \):
- \( x_1 = 97 \), \( y_1 = 29 \)
- \( x_2 = 39 \), \( y_2 = -19 \)

Using the formula:
\[
m_t = \frac{-19 - 29}{39 - 97} = \frac{-48}{-58} = \frac{48}{58} = \frac{24}{29}
\]

**Slope of Line \( u \)**:

For line \( u \), the points are \( (23, 10) \) and \( (98, 2) \):
- \( x_1 = 23 \), \( y_1 = 10 \)
- \( x_2 = 98 \), \( y_2 = 2 \)

Using the formula again:
\[
m_u = \frac{2 - 10}{98 - 23} = \frac{-8}{75}
\]

**Comparison of Slopes**:

1. **Parallel Lines**: Two lines are parallel if their slopes are equal.
2. **Perpendicular Lines**: Two lines are perpendicular if the product of their slopes is \(-1\).

Now, let's check if the lines are parallel or perpendicular.

1. Slopes:
- \( m_t = \frac{24}{29} \)
- \( m_u = \frac{-8}{75} \)

2. Check for parallelism:
- Since \( \frac{24}{29} \neq \frac{-8}{75} \), the lines are not parallel.

3. Check for perpendicularity:
- Calculate the product of the slopes:
\[
m_t \cdot m_u = \left(\frac{24}{29}\right) \cdot \left(\frac{-8}{75}\right) = \frac{24 \cdot -8}{29 \cdot 75} = \frac{-192}{2175}
\]
Since \( \frac{-192}{2175} \neq -1 \), the lines are not perpendicular.

**Conclusion**: Lines \( t \) and \( u \) are neither parallel nor perpendicular.