A line perpendicular to \( A B \) passes through the origin and through the point \( (6, r) \). Find the value of \( r \).

To find the value of \( r \) for the point \( (6, r) \) such that the line passing through the origin and this point is perpendicular to the line segment \( AB \), we need to first determine the slope of the line segment \( AB \). Let’s say the coordinates of points \( A \) and \( B \) are \( A(x_1, y_1) \) and \( B(x_2, y_2) \): 1. The slope of the line \( AB \) is given by the formula: \[ \text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} \] 2. A perpendicular line will have a slope that is the negative reciprocal of the slope of \( AB \). If the slope of \( AB \) is \( m \), the slope \( m_{\perp} \) of the line that is perpendicular to it is: \[ m_{\perp} = -\frac{1}{m} \] 3. The line through the origin and the point \( (6, r) \) has a slope calculated as: \[ \text{slope} = \frac{r - 0}{6 - 0} = \frac{r}{6} \] 4. To ensure the lines are perpendicular, we set the slopes to relate as follows: \[ \frac{r}{6} \cdot m = -1 \] Therefore, \[ r = -\frac{6}{m} \] Without the values of \( m \) (the slope of line segment \( AB \)), we cannot find a specific number for \( r \). If we assume \( AB \) is horizontal (i.e., \( y_2 = y_1 \)), then \( m = 0\), making \( r \) undefined as that would require division by zero. However, if we assign \( m = 1\) as a potential slope, we can have: \[ r = -\frac{6}{1} = -6 \] Thus, if we assume different slopes, we can derive different values of \( r \). For example, if \( m = 2\) then \( r = -3\). The specific line segment needs to be defined to give a proper final value for \( r\). In an exam context, one might find \( r = -6\) under certain assumptions about the relationship between points \( A \) and \( B\). Hence, various conditions can lead to different \( r\) values depending on \( AB\)'s slope.