Problem 45 - Algebra 2

Question

Geometry In the regular polyhedron described below, all faces are congruent polygons. Use a system of three linear equations to find the numbers of vertices, edges, and faces. Every face has five edges and every edge is shared by two faces. Every face has five vertices and every vertex is shared by three faces. The sum of the number of vertices and faces is two more than the number of edges.

Step-by-Step Solution

Answer

The polyhedron has 20 vertices (V), 30 edges (E), and 12 faces (F).

Step 1: Define Variables

Let's define variables V for the number of vertices, E for the number of edges, and F for the number of faces.

Step 2: Set up Equations Based on given information

Use the given geometrical characteristics to establish the following equations: 1. Because each face has 5 edges and each edge is shared by two faces, the relationship can be expressed as: \( E = \frac{5F}{2} \).2. Since every face has 5 vertices and every vertex is shared by three faces, the relationship is: \( V = \frac{5F}{3} \).3. The sum of V and F is two more than the E, which gives us: \( V + F = E + 2 \).

Step 3: Substitute the Expressions for V and E into the Third Equation

Substitute the expressions for V and E into the third equation \( V + F = E + 2 \). This gives us \( \frac{5F}{3} + F = \frac{5F}{2} + 2 \).

Step 4: Solve for F

Solve for F by clearing fractions and combining like terms. First, multiply through by the least common multiple of the denominators, which is 6, to eliminate the fractions, resulting in \( 10F + 6F = 15F + 12 \), which simplifies to \( F = 12 \).

Step 5: Solve for V and E Using the Found Value for F

With F found, substitute F back into the equations to find V and E:1. \( V = \frac{5F}{3} = \frac{5 \times 12}{3} = 20 \)2. \( E = \frac{5F}{2} = \frac{5 \times 12}{2} = 30 \)

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations that share two or more unknowns. These equations are called 'linear' because each term is either a constant or the product of a constant and a single variable. The purpose of the system is to find values for each variable that will satisfy all equations in the system simultaneously.

In our polyhedron problem, we formulated a system with three equations to describe the relationships between the number of vertices (V), edges (E), and faces (F) of the polyhedron. By using the given geometric properties, these equations help us determine the unknown values. Solving such systems typically involves substitution or elimination methods to reduce the system to an equation in one variable, which is exactly what was done in the step-by-step solution provided.

Vertices Edges Faces Relationship

In geometric terms, a polyhedron is a three-dimensional shape with flat faces, straight edges, and sharp corners or vertices. The relationship between vertices, edges, and faces is not random but follows specific patterns.

In the given exercise, two important patterns are highlighted:

• Each face has 5 edges, but each edge is shared by 2 faces, which implies the total number of edges is half the product of the number of faces and 5.
• Each face has 5 vertices, and each vertex is the corner where 3 faces meet, so the total number of vertices is a third of the product of the number of faces and 5.

This relationship can be confusing, but understanding it is essential for solving problems involving polyhedra. This is the same principle used in the Euler's formula, which is another fundamental relationship between the number of vertices, edges, and faces of a polyhedron (V - E + F = 2 for any convex polyhedron).

Geometric Characteristics

Geometric characteristics of shapes like polyhedra include aspects such as symmetry, congruence, and the aforementioned relationships between vertices, edges, and faces. For this exercise, the polyhedron described has congruent faces, meaning they are all the same shape and size—a property that contributes to the polyhedron being regular.

A regular polyhedron is one that is highly symmetrical, not just in face congruence but also in having equal edge lengths, and identical angles between edges. There are only five types of regular polyhedra, famously known as the Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The geometric characteristics provided in the problem are critical to solving for the unknown quantities and to understand the inherent beauty and symmetry of such regular shapes.