Collinearity Check: Distance Formula & Point Identification

Hey guys! Let's dive into a fun geometry problem where we'll use the distance formula to figure out if three points are on the same line and, if they are, pinpoint which one is hanging out in the middle. This is a classic problem that combines algebra and geometry, so buckle up! We will use a friendly and conversational tone to make sure it is easy to understand.

Understanding the Distance Formula and Collinearity

Before we jump into the specific problem, let’s make sure we’re all on the same page with the distance formula and what it means for points to be collinear. Collinearity basically means that points lie on the same straight line. Think of it like lining up beads on a string – if they're all on the same string, they're collinear! The distance formula is our trusty tool for measuring the distance between two points in a coordinate plane. It's derived from the Pythagorean theorem (remember a² + b² = c²?), and it looks like this:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where (x₁, y₁) and (x₂, y₂) are the coordinates of our two points. So, how does this help us with collinearity? Well, if three points A, B, and C are collinear, the sum of the distances between the two pairs of points (AB and BC) should be equal to the distance between the remaining pair of points (AC). In simpler terms, if you add the lengths of the two smaller segments, it should equal the length of the whole segment. This makes intuitive sense, right? If the points aren't in a straight line, this wouldn't hold true because you'd essentially have a triangle, and the sum of any two sides of a triangle is always greater than the third side. To ensure a solid comprehension, it's helpful to consider various examples and counterexamples. For instance, imagine three points that clearly form a triangle; calculating the distances will demonstrate that the sum of any two sides exceeds the third. Conversely, visualizing points neatly aligned on a line reinforces the concept that the sum of the smaller segments equals the entire segment. Understanding this principle is crucial not just for solving mathematical problems but also for grasping fundamental geometric relationships. The distance formula, therefore, isn't just a mathematical tool but a bridge connecting algebraic calculations to geometric visualizations, making complex problems more approachable and understandable. We will explore this concept further in the next section as we tackle our specific problem involving points A, B, and C.

Applying the Distance Formula to Points A, B, and C

Now, let’s get our hands dirty and apply the distance formula to the points given: A(-2, 3), B(2, 1), and C(7, 6). We'll calculate the distances between each pair of points: AB, BC, and AC. This will help us determine if these points are collinear. First, let's calculate the distance between points A and B (AB). Using the formula:

AB = √[(2 - (-2))² + (1 - 3)²] = √[(4)² + (-2)²] = √(16 + 4) = √20

So, the distance AB is √20. Next, we'll find the distance between points B and C (BC):

BC = √[(7 - 2)² + (6 - 1)²] = √[(5)² + (5)²] = √(25 + 25) = √50

Thus, the distance BC is √50. Finally, we calculate the distance between points A and C (AC):

AC = √[(7 - (-2))² + (6 - 3)²] = √[(9)² + (3)²] = √(81 + 9) = √90

So, the distance AC is √90. Now that we have the distances, let's analyze them to see if the points are collinear. Remember, for the points to be collinear, the sum of the two smaller distances should equal the largest distance. In this case, we need to check if AB + BC = AC. Substituting the values we calculated:

√20 + √50 = √90

To simplify this, let's rewrite the square roots by factoring out perfect squares:

√(4 * 5) + √(25 * 2) = √(9 * 10)

This simplifies to:

2√5 + 5√2 = 3√10

At first glance, it's not immediately obvious if this equation holds true. However, if we rewrite √10 as √(5 * 2) or √5 * √2, the right side of the equation becomes 3√5√2. The equation then appears not to balance directly. This is a crucial point because it demonstrates that the simple sum of distances doesn't immediately confirm collinearity when dealing with square roots. We must verify this equality more rigorously, potentially by squaring both sides or converting to decimal approximations to compare values accurately. This step highlights the nuances of working with radicals and the importance of precise calculations when determining collinearity using the distance formula. In the next section, we’ll delve deeper into verifying this equation and definitively answer whether points A, B, and C are indeed collinear.

Verifying Collinearity and Identifying the Intermediate Point

Okay, so we’ve calculated the distances: AB = √20, BC = √50, and AC = √90. We tried adding the two smaller distances (AB + BC) and seeing if they equal the largest distance (AC), but it didn't quite add up in a straightforward way. This doesn't necessarily mean they aren't collinear, it just means we need to dig a little deeper to verify. The previous attempt to directly compare the sum of radical expressions to another radical expression required a more rigorous approach to verification. The initial simplification and comparison of 2√5 + 5√2 to 3√10 did not immediately reveal the relationship due to the nature of adding unlike radicals. To accurately determine if √20 + √50 = √90, we need to employ a more robust method, such as squaring both sides of the equation or using decimal approximations for comparison. Let's try another approach. Instead of directly comparing the radical expressions, we'll square each distance to eliminate the square roots, making the comparison more straightforward. This method leverages the property that if a = b, then a² = b², but it's crucial to remember that the converse is only true for non-negative numbers, which distances always are. So, let's square the distances:

AB² = (√20)² = 20 BC² = (√50)² = 50 AC² = (√90)² = 90

Now, we check if AB² + BC² equals AC²:

20 + 50 = 70

This does not equal 90, indicating an error in our initial assumption or calculation pathway towards proving collinearity directly through the sum of distances. The discrepancy suggests we should reassess our approach or the interpretation of the results obtained so far. Given the straightforward calculations and the clear inequality, the focus should shift towards an alternative method for confirming collinearity, as the direct application of the distance sums did not yield the expected result. To resolve this, we will explore an alternative method to confirm collinearity. A more appropriate condition to verify collinearity using distances directly involves checking if the sum of the lengths of two segments equals the length of the third segment. We initially attempted to verify √20 + √50 = √90, but found it lacking. The correct approach is to ensure the segments add up appropriately: AB + BC must equal AC if B lies between A and C, or similar relations if another point is in the middle. Let’s revisit our original distance values: AB = √20, BC = √50, and AC = √90. Now, let’s express these distances in their simplest radical form to make the comparison clearer:

AB = √(4 * 5) = 2√5 BC = √(25 * 2) = 5√2 AC = √(9 * 10) = 3√10

We must determine if any combination of adding two of these distances results in the third. The critical check involves whether 2√5 + 5√2 equals 3√10. There’s no direct simplification or combination possible because these terms involve different radicals. The equation 2√5 + 5√2 = 3√10 cannot be simplified further to prove equality directly because the radicals are unlike. This means that simply adding the distances in their radical form does not easily lead to verification of collinearity in this case. However, we can reconsider our strategy and recognize that if points A, B, and C are collinear, the sum of the lengths of the two shorter segments should equal the length of the longest segment. In our scenario, this means assessing whether √20 + √50 = √90. To address this accurately, we must either convert these radicals to decimal approximations for comparison or attempt to manipulate them algebraically to find a definitive equality or inequality. Let’s explore another method to definitively determine collinearity, which will provide a clearer answer without relying solely on approximations or complex radical manipulations.

Using Slope to Determine Collinearity

A more straightforward method to determine collinearity is by calculating the slopes between pairs of points. If the points are collinear, the slope between any two pairs of points should be the same. The slope formula is:

Slope (m) = (y₂ - y₁) / (x₂ - x₁)

Let's calculate the slope between points A and B (m_AB):

m_AB = (1 - 3) / (2 - (-2)) = -2 / 4 = -1/2

Now, let's calculate the slope between points B and C (m_BC):

m_BC = (6 - 1) / (7 - 2) = 5 / 5 = 1

Since the slopes m_AB and m_BC are different (-1/2 ≠ 1), the points A, B, and C do not lie on the same line. This definitively answers our question about collinearity. Because the slopes between the point pairs are different, there's no need to identify an intermediate point since they aren't on the same line. This approach provides a clear and concise method for determining collinearity without the complexities involved in simplifying and comparing radical expressions. The fact that the slopes are different immediately tells us that these points form a triangle, not a straight line. In summary, by calculating and comparing the slopes between points A and B, and points B and C, we've conclusively shown that these points are not collinear. This method avoids the complications of working with square roots and provides a direct route to the solution. The difference in slopes clearly indicates that these points cannot lie on a single straight line. Therefore, there is no intermediate point to identify, as the condition of collinearity is not met. This reaffirms the importance of choosing the most efficient method for problem-solving in mathematics. While the distance formula can provide insights, the slope formula offered a more direct and less error-prone way to address the collinearity question in this scenario. Next time you encounter a similar problem, remember that calculating slopes can be a powerful tool in your mathematical toolkit.

Conclusion

So, guys, we tackled a geometry problem using the distance formula and slope formula! We learned that points A(-2, 3), B(2, 1), and C(7, 6) do not lie on the same line because the slopes between the points are different. We initially explored the distance formula but found that comparing the sums of radicals could be tricky. Then, we smartly switched gears and used the slope formula, which gave us a clear and quick answer. Remember, the key to problem-solving is sometimes about choosing the right tool for the job! I hope you found this explanation helpful and a little bit fun. Keep practicing, and you'll become geometry wizards in no time!