Finding The Value Of K For Collinear Points (-5, 1), (5, 5), And (k, 7)

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Collinearity in mathematics refers to the property of points lying on the same straight line. Determining whether points are collinear is a fundamental concept in geometry, with applications ranging from basic coordinate geometry problems to more advanced topics in linear algebra and vector spaces. In this article, we will explore how to find the value of k when given three points—(-5, 1), (5, 5), and (k, 7)—that are collinear. Understanding this concept not only helps in solving specific problems but also provides a solid foundation for understanding linear relationships in mathematics.

Understanding Collinearity

Before diving into the solution, it's crucial to understand what collinearity means. Collinear points are points that lie on the same straight line. In a two-dimensional Cartesian plane, three or more points are collinear if a single straight line can pass through all of them. There are several ways to determine if points are collinear, including using the slope formula, the area of a triangle formula, and vector methods. Each method provides a unique perspective and can be chosen based on the given information and personal preference.

One of the most intuitive ways to check for collinearity is by examining the slopes between pairs of points. If the slope between any two pairs of points is the same, then the points are collinear. The slope (m) between two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

If we calculate the slope between the first two points and the slope between the second and third points, and they are equal, then all three points lie on the same line. This method is straightforward and easy to apply, making it a popular choice for solving collinearity problems. The area of a triangle method is another elegant way to determine collinearity. Three points are collinear if and only if the area of the triangle formed by these points is zero. The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) can be calculated using the formula:

Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

If the calculated area is zero, the points are collinear; otherwise, they are not. This method is particularly useful when dealing with more complex problems or when the coordinates are given in a way that makes the slope method cumbersome. Vector methods provide a more abstract but equally powerful way to determine collinearity. If three points A, B, and C are collinear, then the vectors AB and AC are parallel. This means that one vector is a scalar multiple of the other. We can express the vectors AB and AC as:

AB = B - A AC = C - A

If there exists a scalar t such that AB = t * AC, then the points A, B, and C are collinear. This method is widely used in higher-level mathematics and physics, especially when dealing with three-dimensional space.

Solving for k Using the Slope Method

For the given points (-5, 1), (5, 5), and (k, 7), we will use the slope method to find the value of k. This method involves calculating the slopes between pairs of points and setting them equal to each other.

Step 1: Calculate the Slope Between (-5, 1) and (5, 5)

Let's denote (-5, 1) as (x1, y1) and (5, 5) as (x2, y2). Using the slope formula:

m1 = (y2 - y1) / (x2 - x1) = (5 - 1) / (5 - (-5)) = 4 / 10 = 2/5

So, the slope between the first two points is 2/5.

Step 2: Calculate the Slope Between (5, 5) and (k, 7)

Now, let's denote (5, 5) as (x1, y1) and (k, 7) as (x2, y2). Using the slope formula:

m2 = (y2 - y1) / (x2 - x1) = (7 - 5) / (k - 5) = 2 / (k - 5)

So, the slope between the second and third points is 2 / (k - 5).

Step 3: Set the Slopes Equal and Solve for k

For the points to be collinear, the slopes must be equal. Therefore, we set m1 equal to m2:

2/5 = 2 / (k - 5)

To solve for k, we can cross-multiply:

2 * (k - 5) = 2 * 5 2k - 10 = 10 2k = 20 k = 10

Thus, the value of k that makes the points (-5, 1), (5, 5), and (k, 7) collinear is 10.

Verification Using the Area of a Triangle Method

To further verify our result, we can use the area of a triangle method. If the area of the triangle formed by the points (-5, 1), (5, 5), and (10, 7) is zero, then the points are collinear.

The area of the triangle is given by:

Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Plugging in the coordinates:

Area = 0.5 * |-5(5 - 7) + 5(7 - 1) + 10(1 - 5)| Area = 0.5 * |-5(-2) + 5(6) + 10(-4)| Area = 0.5 * |10 + 30 - 40| Area = 0.5 * |0| Area = 0

Since the area of the triangle is zero, the points (-5, 1), (5, 5), and (10, 7) are indeed collinear. This confirms our result obtained using the slope method.

Alternative Method: Using Determinants

Another elegant method to check for collinearity involves using determinants. Given three points (x1, y1), (x2, y2), and (x3, y3), they are collinear if the determinant of the following matrix is zero:

| x1 y1 1 | | x2 y2 1 | = 0 | x3 y3 1 |

In our case, the points are (-5, 1), (5, 5), and (k, 7). Plugging these into the determinant, we get:

| -5 1 1 | | 5 5 1 | = 0 | k 7 1 |

Expanding the determinant, we have:

-5(5 * 1 - 1 * 7) - 1(5 * 1 - 1 * k) + 1(5 * 7 - 5 * k) = 0 -5(5 - 7) - (5 - k) + (35 - 5k) = 0 -5(-2) - 5 + k + 35 - 5k = 0 10 - 5 + k + 35 - 5k = 0 40 - 4k = 0 4k = 40 k = 10

This method also confirms that the value of k is 10 for the points to be collinear.

Real-World Applications of Collinearity

The concept of collinearity is not just a theoretical exercise in mathematics; it has several real-world applications. Understanding collinearity is crucial in various fields, including:

  1. Navigation and Surveying: In surveying, determining if points are collinear is essential for mapping and land measurement. Surveyors use collinearity to ensure that landmarks and reference points lie on the same line, which is crucial for accurate measurements.
  2. Computer Graphics: In computer graphics, collinearity is used in line drawing algorithms and geometric modeling. For instance, when drawing a straight line on a screen, the algorithm checks for collinearity to ensure that all pixels lie on the same line.
  3. Physics and Engineering: In physics, understanding collinearity is important in mechanics and optics. For example, in optics, the alignment of lenses and mirrors in optical instruments relies on the principle of collinearity.
  4. Astronomy: Astronomers use collinearity to align telescopes and to determine the positions of celestial objects. When observing eclipses or transits, the alignment of the Sun, Earth, and other celestial bodies must be collinear for the event to be visible.
  5. Robotics: In robotics, collinearity is used in path planning and navigation. Robots use sensors to identify collinear points and navigate along straight paths.

These applications highlight the practical significance of collinearity in various disciplines. By understanding the concept and methods for determining collinearity, professionals in these fields can solve complex problems and ensure accuracy in their work.

Conclusion

In this article, we have explored the concept of collinearity and demonstrated how to find the value of k for the points (-5, 1), (5, 5), and (k, 7) to be collinear. We used the slope method as the primary approach and verified our result using the area of a triangle method and determinants. We found that the value of k is 10. Understanding collinearity is not only crucial for solving mathematical problems but also has practical applications in various real-world scenarios, including surveying, computer graphics, physics, astronomy, and robotics. By mastering this concept, you can enhance your problem-solving skills and gain a deeper understanding of linear relationships in mathematics.