Question

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

1. $m_{overleftrightarrow{pq}} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{clubsuit}{c - a}$ $clubsuit = d - b$
2. $m_{overleftrightarrow{pq}} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{c - a}{heartsuit}$ $heartsuit =$ options: $-d - (-b)$, $b - d$, $c - a$

Explanation:

Step1: Identify points for \( \overleftrightarrow{P'Q'} \)

Points \( P'(-b, a) \) and \( Q'(-d, c) \). So \( y_2 = c \), \( y_1 = a \), \( x_2 = -d \), \( x_1 = -b \).

Step2: Calculate \( y_2 - y_1 \) and \( x_2 - x_1 \)

\( y_2 - y_1 = c - a \) (already given for numerator).
\( x_2 - x_1 = -d - (-b) = -d + b = b - d \)? Wait, no—wait, the formula is \( \frac{y_2 - y_1}{x_2 - x_1} \). Wait, the numerator is \( c - a \), so denominator should be \( x_2 - x_1 = -d - (-b) = -d + b = b - d \)? Wait, no, let's check the points: \( P'(-b, a) \), \( Q'(-d, c) \). So \( x_2 - x_1 = -d - (-b) = -d + b = b - d \)? Wait, but the options include \( -d - (-b) \), \( b - d \), \( c - a \). Wait, the denominator \( \heartsuit \) in \( \frac{c - a}{\heartsuit} \) is \( x_2 - x_1 \), which is \( -d - (-b) = -d + b = b - d \)? Wait, no, \( -d - (-b) = -d + b = b - d \), but \( -d - (-b) \) is the expanded form. Wait, let's compute \( x_2 - x_1 \): \( x_2 = -d \), \( x_1 = -b \), so \( x_2 - x_1 = -d - (-b) = -d + b = b - d \). But the first option is \( -d - (-b) \), which is equal to \( b - d \). Wait, but let's see the formula for slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For \( \overleftrightarrow{P'Q'} \), \( y_2 = c \), \( y_1 = a \), so \( y_2 - y_1 = c - a \). \( x_2 = -d \), \( x_1 = -b \), so \( x_2 - x_1 = -d - (-b) = -d + b = b - d \). But the options have \( -d - (-b) \), \( b - d \), \( c - a \). Wait, the denominator is \( x_2 - x_1 \), so it's \( -d - (-b) \) (since \( x_2 = -d \), \( x_1 = -b \), so \( x_2 - x_1 = -d - (-b) \)). Let's verify: \( -d - (-b) = -d + b = b - d \), but the expression \( -d - (-b) \) is the direct calculation from the coordinates. So the denominator \( \heartsuit \) is \( -d - (-b) \).

Wait, no—wait, the slope formula is \( \frac{y_2 - y_1}{x_2 - x_1} \). For \( P'(-b, a) \) and \( Q'(-d, c) \), \( y_2 = c \), \( y_1 = a \), so \( y_2 - y_1 = c - a \). \( x_2 = -d \), \( x_1 = -b \), so \( x_2 - x_1 = -d - (-b) = -d + b = b - d \). But the options are \( -d - (-b) \), \( b - d \), \( c - a \). So \( x_2 - x_1 = -d - (-b) \), which is the same as \( b - d \), but the first option is \( -d - (-b) \), which is the expanded form. So the denominator \( \heartsuit \) is \( -d - (-b) \).

Answer:

\(-d - (-b)\)