There are 3 random points x, y, and z on a coordinate plane. If the absolute value of the slope of xy is the absolute value of the slope of yz and also the absolute value of the slope of zx are the points collinear

Yes, the points \(x\), \(y\), and \(z\) are collinear.

Here's why:

1. The slope of a line between two points on the coordinate plane is calculated as the change in \(y\) divided by the change in \(x\) (i.e., \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\)).
2. If the **absolute values** of the slopes between the pairs of points \(xy\), \(yz\), and \(zx\) are the same, it implies that the lines formed by these points are either parallel or lie along the same straight line.
3. Since all three points share the same absolute slope when connected pairwise, and there are no distinct parallel lines through three distinct points in the plane, the only possibility left is that they lie on the same line.
4. Hence, points \(x\), \(y\), and \(z\) are collinear.

Would you like to go deeper into how to calculate the slopes or any other specific part?

Here are 5 related questions you might want to explore:
1. How do you calculate the slope between two points on a coordinate plane?
2. What does it mean geometrically for three points to be collinear?
3. Can the absolute value of the slopes be the same but the points still not be collinear?
4. How can you prove collinearity using the area of a triangle formed by three points?
5. What is the difference between slopes of parallel and perpendicular lines?

**Tip:** When working with slopes, remember that two lines are parallel if their slopes are exactly equal, but for collinearity, it’s sufficient to have the same slope for all pairs of points.

There are 3 random points x, y, and z on a coordinate plane. If the absolute value of the slope of xy is the absolute value of the slope of yz and also the absolute value of the slope of zx are the points collinear?

Determine if three points on a coordinate plane are collinear by checking if the absolute values of their slopes between each pair are equal. This problem involves concepts of coordinate geometry, slope, and collinearity, suitable for Grades 9-12 students.

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