Question

1. a(0, -2), b(0, 7), c(3, -5), d(6, -5)

there is a table with columns m(\overleftrightarrow{ab}), m(\overleftrightarrow{cd}), types of lines. some handwritten notes and calculations are present around, like \frac{9}{0}, \frac{-10}{3}, \frac{9}{0}, \frac{10}{3}, and neither in the table.

Explanation:

Step1: Calculate slope of \( \overline{AB} \)

The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For \( A(0, -2) \) and \( B(0, 7) \), \( x_1 = 0, y_1 = -2, x_2 = 0, y_2 = 7 \). So \( m(\overline{AB}) = \frac{7 - (-2)}{0 - 0} = \frac{9}{0} \), which is undefined (vertical line).

Step2: Calculate slope of \( \overline{CD} \)

For \( C(3, -5) \) and \( D(6, -5) \), \( x_1 = 3, y_1 = -5, x_2 = 6, y_2 = -5 \). Using slope formula, \( m(\overline{CD}) = \frac{-5 - (-5)}{6 - 3} = \frac{0}{3} = 0 \) (horizontal line).

Step3: Determine line type

A vertical line (undefined slope) and a horizontal line (slope 0) are perpendicular? No, wait: vertical and horizontal lines are perpendicular. Wait, earlier miscalculation? Wait, \( \overline{AB} \) is vertical (x-coordinates same), \( \overline{CD} \) is horizontal (y-coordinates same). Vertical and horizontal lines are perpendicular (slope of vertical is undefined, slope of horizontal is 0, and their product is undefined*0, but geometrically they are perpendicular). Wait, original wrong? Let's recheck.

Wait, \( A(0,-2) \), \( B(0,7) \): x=0, so vertical line. \( C(3,-5) \), \( D(6,-5) \): y=-5, horizontal line. Vertical and horizontal lines are perpendicular (they intersect at 90 degrees). So types of lines: perpendicular.

Wait, let's recalculate slopes:

\( m(\overline{AB}) \): \( \frac{7 - (-2)}{0 - 0} = \frac{9}{0} \) (undefined, vertical line).

\( m(\overline{CD}) \): \( \frac{-5 - (-5)}{6 - 3} = \frac{0}{3} = 0 \) (horizontal line).

Vertical and horizontal lines are perpendicular. So the types of lines are perpendicular.

Answer:

For \( m(\overline{AB}) \): undefined (vertical line), \( m(\overline{CD}) \): 0 (horizontal line), Types of Lines: Perpendicular.

So filling the table:

\( m(\overline{AB}) \): Undefined (or \( \frac{9}{0} \) which is undefined), \( m(\overline{CD}) \): 0 (or \( \frac{0}{3} = 0 \)), Types of Lines: Perpendicular.