Question

Determine whether each statement is true or false. Three noncollinear points determine a plane.

Short answer

True.

Step-by-step solution

Define Collinear and Noncollinear Points

Collinear points are points that lie on the same straight line. Noncollinear points do not all lie on the same straight line, meaning at least one of the points is not on the line that passes through the other two points.

Understand the Definition of a Plane

In geometry, a plane is a flat, two-dimensional surface that extends infinitely in all directions. To define a plane, we generally need at least three points. However, these points must not be collinear; otherwise, they only form a line, not a plane.

Application of the Plane Defining Property

According to geometric principles, three noncollinear points are used to define a plane uniquely. This means if you are given three points and they do not all lie on the same line, you can draw a flat surface (plane) that passes through all of them.

Conclusion on the Statement

Since noncollinear points are essential for determining a specific plane – and three such points can be used to uniquely define a plane as per geometric principles – the given statement is true.

Key concepts

Collinear Points

Collinear points are points that lie on the same straight line. Imagine placing dots on a piece of string; if you can align those dots along the string without bending or looping it back, those dots are collinear. This geometric concept is crucial because it tells us whether the points can be used to define something larger, like a plane.

Key aspects of collinear points include:

• All points lie on a single line.
• If three points are collinear, knowing the location of any two automatically gives you the line.
• They do not help in defining a plane as all points rest along one dimension.

Understanding collinearity helps in visualizing geometric arrangements and forming complex shapes or structures from basic elements.

Noncollinear Points

Noncollinear points, on the other hand, are points that do not all lie on the same straight line. This is where geometry starts to get interesting. Imagine placing three dots on a sheet of paper that don't align no matter how you try to draw a straight line through them. These points are noncollinear.

Here's why noncollinear points are special:

• The presence of a third point not aligned with the first two creates the possibility of a unique plane.
• Noncollinear points form the basis for defining a plane in a two-dimensional space.
• They ensure a structure cannot collapse into a single line.

In geometric terms, noncollinear points are like the anchors marking the borders of your canvas, ensuring that a flat surface, or plane, is realized from their positions.

Plane Definition

A plane in geometry is essentially a flat, two-dimensional surface extending infinitely in all directions. Think of it like an endless sheet of paper or a perfectly flat tabletop that has no edges. This concept is foundational in geometry because it provides a simple but profound way to explore and understand space.

Understanding the principles of plane definition includes:

• A minimum of three noncollinear points is required to determine a plane.
• If the three points are collinear, they cannot define a unique plane; they merely outline a line.
• Through any three noncollinear points, there is exactly one plane.

By realizing the role of planes, students can go beyond the constraints of lines and start exploring the vastness of spaces that these planes can encompass and define.