To find the equation of a line that is perpendicular to the line segment AB and passes through the point (1, 5), we first need to determine the slope of line AB.
Step 1: Find the slope of line segment AB.
The formula for the slope \( m \) between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \( A(6, 4) \) and \( B(9, 6) \):
\[ m_{AB} = \frac{6 - 4}{9 - 6} = \frac{2}{3} \]
Step 2: Find the slope of the perpendicular line.
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, if the slope of line AB is \( \frac{2}{3} \), the slope \( m_p \) of the perpendicular line is:
\[ m_p = -\frac{1}{m_{AB}} = -\frac{1}{\frac{2}{3}} = -\frac{3}{2} \]
Step 3: Use the point-slope form to find the equation of the perpendicular line.
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Substituting the slope \( m_p = -\frac{3}{2} \) and the point \( (1, 5) \):
\[ y - 5 = -\frac{3}{2}(x - 1) \]
Step 4: Simplify the equation.
Distributing the slope on the right side:
\[ y - 5 = -\frac{3}{2}x + \frac{3}{2} \]
Adding 5 to both sides:
\[ y = -\frac{3}{2}x + \frac{3}{2} + 5 \]
Converting 5 into a fraction to combine:
\[ 5 = \frac{10}{2} \]
Thus, we have:
\[ y = -\frac{3}{2}x + \frac{3}{2} + \frac{10}{2} = -\frac{3}{2}x + \frac{13}{2} \]
Conclusion:
The equation of the line that is perpendicular to line segment AB and passes through the point (1, 5) is:
\[ y = -\frac{3}{2}x + \frac{13}{2} \]
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