The line CD which passes through the points C(2,1) and D(6,1) is perpendicular to the line AB which passes through the point B(5,3). Find the equation of the line AB.
Understand the Problem
The question asks to find the equation of a line AB, given that line CD is perpendicular to it, with the coordinates of points C and D known, and that line AB passes through point B. We will first find the slope of line CD, then determine the slope of line AB using the property that perpendicular lines have negative reciprocal slopes. Finally, with the slope of AB and a point on it, we can find its equation.
Answer
$x = 5$
Answer for screen readers
$x = 5$
Steps to Solve
-
Find the slope of line CD The slope of a line given two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$. For line CD, $C(2,1)$ and $D(6,1)$, so the slope $m_{CD}$ is: $m_{CD} = \frac{1 - 1}{6 - 2} = \frac{0}{4} = 0$
-
Find the slope of line AB Since line AB is perpendicular to line CD, its slope $m_{AB}$ is the negative reciprocal of $m_{CD}$. Since $m_{CD} = 0$, line CD is horizontal, so line AB must be vertical. Therefore, the slope of line AB is undefined. A vertical line has an equation of the form $x = c$, where $c$ is a constant.
-
Determine the equation of line AB Since line AB passes through point B(5,3) and is a vertical line, the x-coordinate of every point on the line is 5. So, the equation of line AB is $x = 5$.
$x = 5$
More Information
The line CD is horizontal, and the line AB is vertical.
Tips
A common mistake is to try to apply the negative reciprocal rule when the slope of one line is zero. Remember that the negative reciprocal of 0 is undefined, implying a vertical line.
AI-generated content may contain errors. Please verify critical information
