Finding The Equation Of A Perpendicular Line Triangle ABC

In geometry, understanding the relationships between lines, points, and shapes is fundamental. In this article, we will explore how to find the equation of a line that passes through a specific point and is perpendicular to a given line segment. We will use the example of triangle $ABC$, defined by the points $A(2,9)$, $B(8,4)$, and $C(-3,-2)$, to illustrate this process. Our goal is to determine the equation of a line that passes through point $C$ and is perpendicular to the line segment $\overline{AB}$. This involves several key steps, including calculating the slope of $\overline{AB}$, finding the slope of the perpendicular line, and using the point-slope form to derive the equation of the line. Let's delve into the details and break down each step to achieve our objective.

1. Understanding the Problem

At the heart of this problem is the task of finding a line that satisfies two specific conditions. First, it must pass through the point $C(-3, -2)$. Second, it must be perpendicular to the line segment $\overline{AB}$, which connects points $A(2, 9)$ and $B(8, 4)$. The concept of perpendicularity is crucial here; two lines are perpendicular if they intersect at a right angle (90 degrees). This geometric relationship has a significant algebraic implication: the slopes of perpendicular lines are negative reciprocals of each other. This means that if a line has a slope of $m$, a line perpendicular to it will have a slope of $-1/m$. This property will be instrumental in finding the slope of our desired line. To solve this problem, we will first calculate the slope of $\overline{AB}$. Then, we will determine the negative reciprocal of this slope to find the slope of the line perpendicular to $\overline{AB}$. Finally, we will use the point-slope form of a linear equation, along with the coordinates of point $C$, to write the equation of the line. This step-by-step approach will allow us to systematically find the equation of the line that meets the given conditions. The concepts of slope, perpendicularity, and linear equations are fundamental in coordinate geometry, and mastering these concepts is essential for solving a wide range of geometric problems. Let's proceed with the calculations to find the equation of the line.

2. Calculate the Slope of $\overline{AB}$

The slope of a line is a measure of its steepness and direction. It is defined as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. The formula for the slope ($m$) between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

In our case, the points $A$ and $B$ define the line segment $\overline{AB}$. The coordinates of point $A$ are $(2, 9)$, and the coordinates of point $B$ are $(8, 4)$. Plugging these values into the slope formula, we get:

$m_{AB} = \frac{4 - 9}{8 - 2} = \frac{-5}{6}$

So, the slope of the line segment $\overline{AB}$ is $-\frac{5}{6}$. This negative slope indicates that the line slopes downward from left to right. The magnitude of the slope, $\frac{5}{6}$, tells us how steep the line is; a larger magnitude indicates a steeper line. Now that we have the slope of $\overline{AB}$, we can use the property of perpendicular lines to find the slope of the line perpendicular to it. Remember that the slopes of perpendicular lines are negative reciprocals of each other. This relationship is crucial for the next step in our problem-solving process. By understanding how to calculate the slope between two points, we have laid the groundwork for finding the equation of the line perpendicular to $\overline{AB}$. This is a fundamental concept in coordinate geometry and is essential for solving various geometric problems involving lines and their relationships.

3. Determine the Slope of the Perpendicular Line

As we discussed earlier, two lines are perpendicular if they intersect at a right angle, and their slopes are negative reciprocals of each other. This means that if one line has a slope of $m$, a line perpendicular to it will have a slope of $-1/m$. We have already calculated the slope of $\overline{AB}$ to be $-\frac{5}{6}$. To find the slope of the line perpendicular to $\overline{AB}$, we need to take the negative reciprocal of $-\frac{5}{6}$.

The negative reciprocal of a fraction is found by first changing the sign of the fraction and then inverting it (swapping the numerator and denominator). So, the negative reciprocal of $-\frac{5}{6}$ is:

$- \frac{1}{(-\frac{5}{6})} = \frac{6}{5}$

Therefore, the slope of the line perpendicular to $\overline{AB}$ is $\frac{6}{5}$. This positive slope indicates that the line slopes upward from left to right. The fact that the slope is $\frac{6}{5}$ tells us that for every 5 units we move to the right along the line, we move 6 units up. This slope is essential for defining the direction of the line we are trying to find. Now that we have the slope of the perpendicular line, we also know that this line must pass through point $C(-3, -2)$. With both the slope and a point on the line, we have enough information to write the equation of the line using the point-slope form. The concept of negative reciprocals is a cornerstone of understanding perpendicularity in coordinate geometry, and it allows us to relate the slopes of lines that intersect at right angles. This understanding is crucial for solving a wide range of geometric problems.

4. Use the Point-Slope Form to Find the Equation

The point-slope form is a convenient way to express the equation of a line when you know a point on the line and its slope. The point-slope form of a linear equation is given by:

$y - y_1 = m(x - x_1)$

where:

• $(x_1, y_1)$ is a point on the line,
• $m$ is the slope of the line.

We know that the line we are trying to find passes through point $C(-3, -2)$ and has a slope of $\frac{6}{5}$. Plugging these values into the point-slope form, we get:

$y - (-2) = \frac{6}{5}(x - (-3))$

Simplifying this equation, we have:

$y + 2 = \frac{6}{5}(x + 3)$

This is the point-slope form of the equation of the line. To get the equation in the slope-intercept form ($y = mx + b$), we need to distribute the $\frac{6}{5}$ and isolate $y$:

$y + 2 = \frac{6}{5}x + \frac{18}{5}$

Subtracting 2 from both sides (and noting that $2 = \frac{10}{5}$), we get:

$y = \frac{6}{5}x + \frac{18}{5} - \frac{10}{5}$

$y = \frac{6}{5}x + \frac{8}{5}$

Thus, the equation of the line passing through point $C$ and perpendicular to $\overline{AB}$ is $y = \frac{6}{5}x + \frac{8}{5}$. This equation tells us everything we need to know about the line: its slope is $\frac{6}{5}$, and its y-intercept is $\frac{8}{5}$. The point-slope form is a powerful tool for finding the equation of a line, and it is particularly useful when you have a point and the slope. By understanding and applying the point-slope form, we can easily derive the equation of a line that meets specific conditions.

5. Final Equation and Conclusion

After performing the calculations, we have successfully determined the equation of the line that passes through point $C(-3, -2)$ and is perpendicular to the line segment $\overline{AB}$ with endpoints $A(2, 9)$ and $B(8, 4)$. The final equation of the line is:

$y = \frac{6}{5}x + \frac{8}{5}$

This equation is in slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In this case, the slope of the line is $\frac{6}{5}$, and the y-intercept is $\frac{8}{5}$. This means that the line intersects the y-axis at the point $(0, \frac{8}{5})$. The positive slope indicates that the line rises from left to right, and for every 5 units you move to the right along the line, you move 6 units up. To summarize, we started by understanding the problem and identifying the key concepts: perpendicularity, slope, and the equation of a line. We then calculated the slope of $\overline{AB}$ using the coordinates of points $A$ and $B$. Next, we found the slope of the line perpendicular to $\overline{AB}$ by taking the negative reciprocal of the slope of $\overline{AB}$. Finally, we used the point-slope form of a linear equation and the coordinates of point $C$ to derive the equation of the line. This step-by-step process allowed us to systematically solve the problem and find the equation of the line that meets the given conditions. Understanding these concepts and techniques is crucial for solving a wide range of geometry problems and for further studies in mathematics and related fields.

In conclusion, the completed equation for a line passing through point C and perpendicular to $\overline{AB}$ is:

$y = \frac{6}{5}x + \frac{8}{5}$