Question

Determine whether the points are collinear. (Three points are collinear if they lie on the same line.) $$ (0,4),(7,-6),(-5,11) $$

Short answer

The points (0,4), (7,-6), and (-5,11) are not collinear.

Step-by-step solution

Calculate the slopes.

First, calculate the slope of the line segment between the first two points. Let's denote the first point as A(0,4) and the second as B(7,-6). Using the slope formula, we get: \[ Slope_{AB} = \frac{-6-4}{7-0} = \frac{-10}{7} \]. Next, find the slope of the line segment between the second and the third points. Let's denote the second point as B(7,-6) and the third as C(-5,11). Applying the slope formula again, we obtain: \[ Slope_{BC} = \frac{11-(-6)}{-5-7} = \frac{17}{-12} = -\frac{17}{12} \].

Compare the slopes.

In this step, compare the two slopes calculated in step 1. If the slopes are equal, then the points are collinear. If not, the points are not collinear.

Conclusion

Examining the slopes, it is clear that \( Slope_{AB} \) is not equal to \( Slope_{BC} \), which means that the points (0,4), (7,-6), and (-5,11) are not collinear.

Key concepts

Slope Formula

The slope formula is a fundamental tool in geometry that helps determine the steepness of a line segment connecting two points. It is expressed as the ratio of the 'rise' (the difference in the y-coordinates) over the 'run' (the difference in the x-coordinates). In mathematical terms, given two points, A(x1, y1) and B(x2, y2), the slope (m) is calculated using this formula:
\[\begin{equation} m = \frac{y2 - y1}{x2 - x1}\end{equation}\]
Understanding this formula is crucial when dealing with collinear points, as collinearity requires that the slopes of any two pairs of points be equal. Consequently, if you have three points, and you find two distinct slopes using pairs of these points, and they are not equal, it infers that these points are not lying on the same straight line, hence, not collinear.

Line Segment

A line segment in geometry is a part of a line that is bounded by two distinct end points. It contains these end points and all the points on the line between them. Line segments are often used to connect points on a graph to construct shapes and determine geometric properties like slope and distance.

Understanding Collinearity with Line Segments

To ascertain whether points are collinear, you can visualize or plot these points, draw line segments between them, and assess if they all lie on a single straight line. The exercise of calculating the slope becomes a practical approach to confirm this visually assessed collinearity mathematically, by checking if the slope of the line segments formed by each pair of points is identical.

Calculus in Geometry

The principles of calculus often intersect with geometric concepts, providing a powerful framework for analyzing various properties within geometric shapes. While calculus may seem overwhelming, its basic principles, such as derivatives and integrals, are linked to the slopes and areas within geometry.
For example, the derivative of a function at a point can be understood as the slope of the tangent line to the function's graph at that point. When dealing with less complex geometric tasks such as finding if points are collinear, calculus may not be necessary. However, the knowledge of calculus becomes essential in more intricate geometric situations, such as finding the rate of change of a curve or the area under a curve—situations where simple algebraic formulas for slope and area are not sufficient.