Proving Collinearity of Points: A Comprehensive Guide

Understanding Collinearity: Collinear points in a plane are points that lie on the same straight line. Determining whether points are collinear is an essential skill in geometry and computer science. This guide will explore various methods to prove collinearity, including the slope method and the triangle area method.

Proving Collinearity Using the Slope Method

In the context of proving collinearity, one of the most straightforward methods involves calculating the slopes between pairs of points. If the slopes are equal, then the points are collinear.

Calculation Steps for Slope Method

Step 1: Calculate the slope between the first two points (3, 4) and (7, 7). The formula for the slope ( m ) between two points ((x_1, y_1)) and ((x_2, y_2)) is: [frac{y_2 - y_1}{x_2 - x_1}] Apply the slope formula: (m_1 frac{7 - 4}{7 - 3} frac{3}{4}) Step 2: Calculate the slope between the second and third points (7, 7) and (11, 10) Apply the slope formula again: (m_2 frac{10 - 7}{11 - 7} frac{3}{4}) Step 3: Calculate the slope between the first and third points (3, 4) and (11, 10) Apply the slope formula once more: (m_3 frac{10 - 4}{11 - 3} frac{6}{8} frac{3}{4}) Conclusion: Since all three slopes ((m_1), (m_2), and (m_3)) are equal ((frac{3}{4})), the points (3, 4), (7, 7), and (11, 10) are collinear.

Proving Collinearity Using the Triangle Area Method

Another method to prove collinearity involves the concept of the area of a triangle formed by the points. If the area is zero, then the points are collinear, as they form a straight line.

Calculation Steps for Triangle Area Method

Step 1: Calculate the lengths of the sides of the triangle Length between (3, 4) and (7, 7): Distance formula: (sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2}) Applying the distance formula:(d_1 sqrt{(7 - 3)^2 (7 - 4)^2} sqrt{4^2 3^2} sqrt{25} 5) Length between (3, 4) and (11, 10): (d_2 sqrt{(11 - 3)^2 (10 - 4)^2} sqrt{8^2 6^2} sqrt{100} 10) Length between (7, 7) and (11, 10): (d_3 sqrt{(11 - 7)^2 (10 - 7)^2} sqrt{4^2 3^2} sqrt{25} 5) Step 2: Calculate the semi-perimeter (s) (s frac{d_1 d_2 d_3}{2} frac{5 10 5}{2} 10) Step 3: Apply Heron’s formula to find the area (A sqrt{s(s - d_1)(s - d_2)(s - d_3)} sqrt{10(10 - 5)(10 - 10)(10 - 5)}) Since one of the terms inside the square root is zero, the area (A 0). Conclusion: A zero area means the points form a straight line, thus proving collinearity.

Summary

In both methods discussed, we utilized mathematical concepts to prove that the points (3, 4), (7, 7), and (11, 10) are collinear. The slope method involves equal slopes between pairs of points, while the triangle area method confirms collinearity by showing a zero area for the triangle formed by the points.

These techniques can be applied to any set of three ordered pairs on a coordinate plane to determine collinearity. If you have any further questions or need additional examples, feel free to ask!