Question

Draw plane \(A B C\) and plane \(D E F\) that intersect in \(\overline{G H}\).

Short answer

Draw two intersecting parallelograms with a common line \( \overline{GH} \).

Step-by-step solution

Draw the First Plane ABC

Start by sketching a plane which we will call plane \( ABC \). To represent a plane on paper, draw a parallelogram. Within this parallelogram, label three non-collinear points as \( A \), \( B \), and \( C \).

Draw the Intersection Line GH

Next, draw a straight line across the plane \( ABC \) which will represent the line of intersection \( \overline{GH} \). Make sure this line is a bit longer than the sides of your parallelogram to show that it extends outside of plane \( ABC \) as well. Label the endpoints on this line as \( G \) and \( H \).

Draw the Second Plane DEF

Now, draw another parallelogram that crosses the line \( \overline{GH} \). This new parallelogram represents plane \( DEF \). Like the first plane, label three non-collinear points within the second parallelogram as \( D \), \( E \), and \( F \).

Highlight the Intersection of Planes

Emphasize that line \( \overline{GH} \) is where the two planes intersect by ensuring it clearly crosses both planes \( ABC \) and \( DEF \). You might want to darken this line to show its importance as the shared intersection line between the two planes.

Key concepts

Geometry Problem Solving

In geometry, problem-solving often involves visualizing spatial relationships, particularly when dealing with intersecting planes. Understanding the intersection of planes is crucial, as it introduces concepts that appear in more complex geometrical problems later on. When two planes intersect in 3D space, they do so along a line shared by both planes, called the line of intersection. This exercise reinforces the understanding of this concept by having you draw two intersecting planes.

Approaching this problem involves tackling a few key steps:

• Recognize that each plane can be represented by a parallelogram.
• Understand that the non-collinear points define each plane within the parallelogram.
• Comprehend that the line of intersection is where the two planes meet.

As you work through this, consider how each part connects, and visualize how they exist not only on the page but also in real 3D space. By grasping these concepts, you set a solid foundation for more challenging geometry problems.

Drawing Geometric Figures

Creating accurate geometric figures is essential for representing concepts such as intersecting planes. Let's explore how to sketch two such planes:

• The first step is to draw a parallelogram labeled as plane \( ABC \). This shape serves as the framework for the plane and encapsulates three non-collinear points \( A \), \( B \), and \( C \).
• Next, draw line \( \overline{GH} \) across this parallelogram. Line \( \overline{GH} \) extends beyond the edges of \( ABC \), depicting an infinite line of intersection.
• Then create a second parallelogram for plane \( DEF \). Ensure it intersects line \( \overline{GH} \) to properly convey the intersection.
• Finally, label non-collinear points \( D \), \( E \), and \( F \) within the second plane.

Consistency in your drawing helps convey the intended geometric relationships. Employing clean lines and accurate labels ensures clarity and understanding for viewers examining your geometric figures.

Non-Collinear Points

Non-collinear points play a pivotal role in defining planes in geometry. It’s crucial to understand this concept when dealing with the intersection of two planes.

A plane is defined by at least three points. However, these points must be non-collinear for the plane to exist as a flat surface rather than a line.

• A set of non-collinear points does not lie on the same straight line. In our scenario, points \( A \), \( B \), and \( C \) in plane \( ABC \) are non-collinear, as are points \( D \), \( E \), and \( F \) in plane \( DEF \).
• This non-collinearity ensures that each set of points creates a distinct plane.

When sketching these planes, always check to ensure that your chosen points do not align on a single straight line. Practicing the recognition and use of non-collinear points will strengthen your spatial reasoning and improve your ability to solve more complex geometry problems.