Question

given: (overleftrightarrow{pq} perp overleftrightarrow{pq})
prove: (left( m_{overleftrightarrow{pq}}
ight) left( m_{overleftrightarrow{pq}}
ight) = -1)

1. (m_{overleftrightarrow{pq}} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\boldsymbol{\text{?}}}{c - a}) (\boldsymbol{\text{?}} =) dropdown options: (d - b), (b - d), (a - c)

check
diagram: coordinate plane with (p(a, b)), (p(-b, a)), (q(c, d)), (q(-d, c)) plotted

Explanation:

Step1: Identify coordinates of P and Q

Point \( P \) is \( (a, b) \) and point \( Q \) is \( (c, d) \). The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For line \( \overleftrightarrow{PQ} \), \( y_2 = d \), \( y_1 = b \), \( x_2 = c \), \( x_1 = a \).

Step2: Apply slope formula

Substitute into the slope formula: \( m_{\overleftrightarrow{PQ}} = \frac{d - b}{c - a} \). So the numerator (the \( \boldsymbol{\frac{y_2 - y_1}{x_2 - x_1}} \) numerator) is \( d - b \).

Answer:

\( d - b \)