Question

if points \(D, E\), and \(F\) are collinear and \(E\) is not between \(D\) and \(F\), then \(F\) is between \(D\) and \(E\).

Short answer

The point \(F\) is between \(D\) and \(E\) since \(E\) is not between \(D\) and \(F\).

Step-by-step solution

Understanding Collinearity

Points are collinear if they lie on the same straight line. Since points \(D, E,\) and \(F\) are collinear, this means they all lie on the same line.

Analyzing the Given Condition

We are given that \(E\) is not between \(D\) and \(F\). This statement tells us that \(E\) does not lie in the segment with endpoints \(D\) and \(F\).

Conclusion Based on Placement

Since \(E\) is not between \(D\) and \(F\), and all points are on a line, \(D\) must be at one endpoint, \(E\) at the other, with \(F\) positioned between them. Thus, \(F\) is the point between \(D\) and \(E\).

Key concepts

Geometry

Geometry is a branch of mathematics that studies the properties and relationships of points, lines, surfaces, and shapes. In this context, understanding these elements is crucial for solving problems related to collinear points. Collinearity indicates that multiple points share the same straight path or line. This concept can help determine the relative positioning of the points in question. To solve problems involving geometry, it’s important to visualize how these points align and the implications of their arrangement. For instance, collinear points are essential in defining line segments and understanding the relationship between multiple points within geometrical arrangements.

Line Segments

A line segment is a part of a line bounded by two distinct endpoints. In geometry, understanding line segments is fundamental. They have a fixed length and contain all points lying between the two endpoints. For example, if you have a line segment between points \(D\) and \(F\), any point that lies direct between \(D\) and \(F\) on the line is said to be part of the segment. In the given problem, we know that point \(E\) is not between \(D\) and \(F\), meaning it doesn't exist on the line segment connecting these two points. Recognizing these distinctions allows us to derive further conclusions about the configuration of the points.

Point Placement

Point placement is crucial in understanding spatial relationships in geometry. It involves analyzing where a point lies relative to others on a line or plane. In the given problem, knowing that \(E\) is not between \(D\) and \(F\) directs us to reconsider how the points are placed. Since all three points are collinear, they must form a specific arrangement on their line. If one point isn't between two others, it influences the possible configurations of these points, for example, showing that \(F\) is between \(D\) and \(E\).

• Visualize the line with points labeled \(D\), \(F\), and \(E\).
• Consider how changing the placement of one point affects the others.
• Understand that this placement helps solve geometric problems involving linear arrangements.

These insights into point placement empower you to approach geometrical problems strategically, leading to accurate solutions.