91Ó°ÊÓ

To determine if three points are collinear, we start by calculating the slope. The slope is a measure of the steepness or incline of a line between two points. This can be done using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)Where

• \( (x_1, y_1) \) are the coordinates of the first point,
• \( (x_2, y_2) \) are the coordinates of the second point.

In our exercise, let's calculate the slope between the first two points, \( (-1, 4) \) and \( (-2, -1) \). Using the formula, we get:\( m = \frac{-1 - 4}{-2 - (-1)} = \frac{-5}{-1} = 5 \) This means the slope between these two points is 5. Similarly, we calculate the slope between the second and third points, \( (-2, -1) \) and \( (1, 14) \):\( m = \frac{14 - (-1)}{1 - (-2)} = \frac{15}{3} = 5 \) Once again, the slope is 5. By comparing these slopes, we can move on to check collinearity.

Analytical geometry is a branch of geometry that uses algebraic equations to describe geometric principles. One important concept in analytical geometry is the slope of a line.

• It allows us to understand the relationship between different points on a plane.
• By calculating the slope between two points, we can determine if they lie on the same line.

In our exercise using analytical geometry, we start with the coordinates of three points: \( (-1, 4), (-2, -1) \), and \( (1, 14) \). By calculating the slopes between these points using the slope formula, we can determine if these three points are collinear. Analytical geometry provides a structured method to solve geometrical problems using algebraic techniques.

Collinearity refers to the property of points lying on the same straight line. To test collinearity using slopes, it is essential to establish a constant slope between each pair of points.

• If the slopes between each pair are equal, the points lie on the same line.
• If they differ, the points are not collinear.

In our exercise, we calculated the slopes between the point pairs \( (-1, 4) \) and \( (-2, -1) \), and \( (-2, -1) \) and \( (1, 14) \). Both slopes turned out to be 5, confirming that these points are collinear. Thus, understanding collinearity helps us verify the alignment of points on a plane through simple slope calculations.