Question

slope of line tg, slope of line cz, point - slope form of line cz; $m_1$, $m_2$, $y - y_1 = m(x - x_1)$; $-\frac{2}{3}$, dropdown, blank

Explanation:

To determine the slope of line \( CZ \) (\( m_2 \)) when we know the slope of line \( TG \) (\( m_1 = -\frac{2}{3} \)), we need to consider the relationship between the two lines. If the lines are perpendicular, the slopes are negative reciprocals of each other. If they are parallel, the slopes are equal.

Step 1: Assume the lines are perpendicular (common relationship for such problems)

The formula for the slope of a line perpendicular to a line with slope \( m \) is \( m_{\perp} = \frac{-1}{m} \).

Given \( m_1 = -\frac{2}{3} \), the negative reciprocal is:
\[
m_2 = \frac{-1}{-\frac{2}{3}} = \frac{3}{2}
\]

Step 2: If the lines are parallel

If the lines are parallel, then \( m_2 = m_1 = -\frac{2}{3} \).

However, since the problem likely involves perpendicular lines (a common scenario), the most probable slope for \( CZ \) (if perpendicular to \( TG \)) is \( \frac{3}{2} \).

Answer:

If perpendicular: \( \boldsymbol{\frac{3}{2}} \)
If parallel: \( \boldsymbol{-\frac{2}{3}} \)

(Assuming perpendicular, the slope of \( CZ \) is \( \frac{3}{2} \).)