Problem 79 - Algebra 2

Question

Points that are on the same line are collinear. Use the definition of slope to determine whether the given points are collinear. $$ (-2,6),(0,2),(1,0) $$

Step-by-Step Solution

Answer

The points are collinear because the slope between every two points is equal.

Step 1: Find the Slope Between the First Two Points

The slope of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Calculate the slope between points \( (-2, 6) \) and \( (0, 2) \) using their coordinates.

Step 2: Find the Slope Between the Next Two Points

Similarly, calculate the slope between points \( (0, 2) \) and \( (1, 0) \) using the same slope formula.

Step 3: Compare the Slopes

If the slopes from Step 1 and Step 2 are equal, the points are collinear because they lie on the same line. If the slopes are different, then the points are not collinear.

Key Concepts

Slope of a Line

Understanding the slope of a line is essential in coordinate geometry as it describes the steepness and the direction of the line. The slope is commonly denoted as 'm' and is a numerical value that results from the ratio of the vertical change ('rise') to the horizontal change ('run') between two distinct points on the line.

When considering any two points, say \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line, we create a right triangle by drawing a horizontal line from the first point to a point directly above or below the second point, and then a vertical line to the second point. The slope is then the vertical side divided by the horizontal side of this right triangle.

If the line goes upwards from left to right, the slope is positive; if it goes downwards, the slope is negative. A horizontal line has a slope of 0 because there is no vertical change, while a vertical line's slope is undefined as there is no horizontal change, which would lead to division by zero in the slope formula.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method combines algebra and geometry to describe the position and relation of geometric figures on a plane. Coordinate geometry is fundamental in determining the collinearity of points, the shapes of figures, and the equations of lines.

For example, by using coordinates, we can easily derive the slope of a line, find the distance between points, and identify geometric properties such as parallelism or perpendicularity of lines. In the context of collinear points, coordinate geometry provides an algebraic means to determine whether multiple points lie on the same line, which opens the door to solving various applied mathematics and physics problems.

Slope Formula

The slope formula is a straightforward expression that allows one to calculate the slope of a line segment between two points. The formula is given by:
\br>\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. It's important to be consistent with the order of subtraction to ensure accurate results. For instance, if you start with \( y_2 \) in the numerator, you must start with \( x_2 \) in the denominator. The slope formula provides a quantitative measure of the inclination of the line and is pivotal in numerous areas of mathematics, from basic algebra to calculus.

When evaluating the collinearity of points, one simply computes the slope between each pair of points; if all the slopes are equal, then all the points are collinear, affirming that they lie on the same straight line.