91Ó°ÊÓ

Let's start by understanding what **slope** means. The slope measures how steep a line is. You can think of it as the rate of change or 'rise over run'. To calculate the slope between any two points \((x_1, y_1)\) and \((x_2, y_2)\), use this formula:
\(\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}\).

In our example, we first calculate the slope between points \((7, -4)\) and \((3, -1)\). The calculation is:
\(\frac{-1 - (-4)}{3 - 7} = \frac{3}{-4} = -\frac{3}{4}\).
This formula helps us understand how point \((7, -4)\) changes as it moves to \((3, -1)\).
Next, we calculate the slope between \((3, -1)\) and \((-2, 2.75)\):
\(\frac{2.75 - (-1)}{-2 - 3} = \frac{3.75}{-5} = -\frac{3.75}{5} = -\frac{3}{4}\).
By comparing these slopes, we confirm that both have a value of -\(\frac{3}{4}\).

Now, let's talk about the triangular area concept. Three points will form a triangle if they are not collinear. However, collinear points will form a triangle with zero area.

To confirm collinearity using the area method, we can leverage the formula for the area of a triangle given three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[ \text{Area} = \frac{1}{2} \big| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \big| \].

If the area equals zero, the points are collinear. Applying this formula to points \((7,-4)\), \((3,-1)\), and \((-2, 2.75)\), we would indeed find the area to be zero, confirming their collinearity through another method.

So, what are **collinear points**? In simple terms, points that lie on the same straight line are collinear. This concept is vital in geometry and algebra.

To determine if points are collinear, we often check if the slopes between each pair of points are equal. If the slopes match, it means there's no change in direction between the points, and they must lie on the same line.

In our example, we used slope calculation to find that the slopes between points \((7, -4)\) and \((3, -1)\) and between \((3, -1)\) and \((-2, 2.75)\) were identical. Thus, we proved that these points are collinear. Another way to assess collinearity is by checking the area of the triangle formed by the points. If it's zero, the points lie on the same line.

Remember to apply either method: comparing slopes or calculating triangular area. Both approaches should yield the same result when accurately performed.