Finding Altitude Equations Of A Triangle With Vertices (-5,6), (-1,-4), And (3,2)

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This article delves into the process of determining the equations of the altitudes within a triangle defined by the vertices (-5, 6), (-1, -4), and (3, 2). Altitudes, in geometrical terms, are perpendicular lines drawn from a vertex of a triangle to the opposite side. Calculating these altitudes involves several key concepts from coordinate geometry, including finding slopes, using the point-slope form of a line, and solving systems of equations. This exploration not only reinforces these fundamental mathematical principles but also provides a practical application in understanding the properties of triangles and their geometric attributes. The step-by-step approach outlined in this article is designed to offer a clear and comprehensive guide, beneficial for students, educators, and anyone with an interest in geometry and analytical problem-solving.

Understanding Altitudes and Their Properties

Before we jump into the calculations, let's solidify our understanding of altitudes in a triangle. An altitude is a line segment drawn from a vertex of the triangle perpendicular to the opposite side (or the extension of the opposite side). Every triangle has three altitudes, one from each vertex. The point where the three altitudes intersect is known as the orthocenter of the triangle. The orthocenter is a crucial point of concurrency within the triangle, providing valuable information about its overall geometry and characteristics. Understanding the properties of altitudes is essential for various applications, from basic geometric proofs to more advanced topics like triangle centers and their relationships. The altitude's perpendicularity to the base is the key to its unique properties and its role in determining the area of the triangle. This perpendicularity also dictates the methods we use to calculate the equations of the altitudes, primarily involving the negative reciprocal of slopes and the point-slope form of linear equations. In essence, altitudes serve as a bridge connecting the vertices and sides of a triangle, allowing us to explore its internal structure and spatial relationships more comprehensively. This foundational knowledge is vital as we proceed with the step-by-step process of finding the equations of the altitudes for the given triangle.

Step 1: Calculate the Slopes of the Sides

To begin, our initial task is to calculate the slopes of the three sides of the triangle. Let's denote the vertices as A(-5, 6), B(-1, -4), and C(3, 2). The slope of a line segment connecting two points (x1, y1) and (x2, y2) is given by the formula: m = (y2 - y1) / (x2 - x1). Applying this formula, we first find the slope of side AB: m_AB = (-4 - 6) / (-1 - (-5)) = -10 / 4 = -5/2. Next, we calculate the slope of side BC: m_BC = (2 - (-4)) / (3 - (-1)) = 6 / 4 = 3/2. Lastly, we find the slope of side CA: m_CA = (6 - 2) / (-5 - 3) = 4 / -8 = -1/2. These slopes are crucial as they will help us determine the slopes of the altitudes, which are perpendicular to these sides. The concept of perpendicular lines is central to our calculations, as the slopes of perpendicular lines are negative reciprocals of each other. By finding the slopes of the sides, we lay the groundwork for determining the slopes of the altitudes and subsequently their equations. This step is not just about applying a formula; it's about understanding the geometric relationships between the sides and altitudes of the triangle, paving the way for a complete solution.

Step 2: Determine the Slopes of the Altitudes

Now that we have the slopes of the sides, we can proceed to determine the slopes of the altitudes. Recall that an altitude is perpendicular to the side it intersects. The slopes of perpendicular lines are negative reciprocals of each other. This means that if the slope of a side is 'm', the slope of the altitude to that side will be '-1/m'. Let's find the slope of the altitude from vertex C to side AB. Since the slope of AB (m_AB) is -5/2, the slope of the altitude from C (m_altitude_C) will be the negative reciprocal of -5/2, which is 2/5. Next, we determine the slope of the altitude from vertex A to side BC. The slope of BC (m_BC) is 3/2, so the slope of the altitude from A (m_altitude_A) will be -2/3. Finally, we find the slope of the altitude from vertex B to side CA. The slope of CA (m_CA) is -1/2, so the slope of the altitude from B (m_altitude_B) will be the negative reciprocal of -1/2, which is 2. These calculated slopes of the altitudes are vital for the next step, where we will use the point-slope form to derive the equations of these lines. Understanding the relationship between the slopes of perpendicular lines is fundamental in this process, ensuring accurate determination of the altitude slopes and, subsequently, their equations.

Step 3: Use the Point-Slope Form to Find the Equations of the Altitudes

With the slopes of the altitudes now known, we can use the point-slope form of a linear equation to find their equations. The point-slope form is given by y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Let's start with the altitude from vertex C(3, 2) to side AB. We already found the slope of this altitude to be 2/5. Using the point-slope form, the equation of the altitude from C is: y - 2 = (2/5)(x - 3). Simplifying this equation, we get: y = (2/5)x - 6/5 + 2, which further simplifies to y = (2/5)x + 4/5. Next, consider the altitude from vertex A(-5, 6) to side BC. The slope of this altitude is -2/3. Applying the point-slope form, the equation of the altitude from A is: y - 6 = (-2/3)(x - (-5)). Simplifying, we have: y = (-2/3)x - 10/3 + 6, which simplifies to y = (-2/3)x + 8/3. Lastly, for the altitude from vertex B(-1, -4) to side CA, the slope is 2. The equation using the point-slope form is: y - (-4) = 2(x - (-1)). Simplifying, we get: y + 4 = 2x + 2, which results in y = 2x - 2. These three equations represent the altitudes of the triangle. This step demonstrates how the point-slope form is a powerful tool for determining the equation of a line when we know a point on the line and its slope. By systematically applying this form to each altitude, we successfully derive their respective equations, completing the primary objective of this problem.

Step 4: Verify the Altitudes (Optional)

While not strictly necessary, verifying the calculated altitudes can be a beneficial step to ensure accuracy. This verification can be done in a couple of ways. One method involves checking if the altitudes are indeed perpendicular to their respective sides. We can do this by confirming that the product of the slopes of a side and its corresponding altitude is -1. For example, the slope of side AB is -5/2, and the slope of the altitude from C is 2/5. Their product is (-5/2) * (2/5) = -1, confirming their perpendicularity. Similarly, we can check the other altitudes and sides. Another verification method involves graphical representation. If you plot the triangle and the calculated altitudes on a graph, you can visually confirm that the altitudes appear to be perpendicular to their respective sides and that they intersect at a single point (the orthocenter). This visual check can help catch any potential errors in calculations or equation derivations. Furthermore, we can solve any two altitude equations simultaneously to find the coordinates of the orthocenter. Then, we can substitute these coordinates into the third altitude equation. If the equation holds true, it confirms that all three altitudes intersect at the same point, lending further credence to our calculations. This step, although optional, adds a layer of confidence to our solution, ensuring that the derived equations accurately represent the altitudes of the triangle. By employing either algebraic checks or graphical verification, we solidify our understanding and the correctness of the solution.

Conclusion

In conclusion, we have successfully determined the equations of the altitudes of the triangle with vertices (-5, 6), (-1, -4), and (3, 2). By systematically applying the principles of coordinate geometry, including calculating slopes, using the point-slope form, and understanding the relationship between perpendicular lines, we derived the equations for each altitude. The equations of the altitudes are: 1. Altitude from C: y = (2/5)x + 4/5 2. Altitude from A: y = (-2/3)x + 8/3 3. Altitude from B: y = 2x - 2. This process not only provides a solution to the specific problem but also reinforces key concepts in geometry and algebra. The step-by-step approach used here can be applied to similar problems involving triangles and their properties. Understanding altitudes is crucial in various mathematical contexts, from basic geometry to more advanced topics like triangle centers and their relationships. The ability to calculate and represent these altitudes algebraically enhances our understanding of triangular geometry and its applications. By following this structured method, one can confidently tackle problems involving altitudes and other geometric properties, strengthening their mathematical problem-solving skills. The exercise of finding altitudes is a testament to the power of analytical geometry in bridging the visual aspects of geometry with the precision of algebraic equations.