Question

There is a relationship between the number of vertices in a polygon and the number of triangles in any triangulation of that polygon. State this relationship and prove it by induction.

Short answer

The relationship is \(t = n - 2\) and it holds true by induction for any polygon with \(n\) vertices.

Step-by-step solution

Understanding the Relationship

The relationship between the number of vertices \(n\) in a polygon and the number of triangles \(t\) in any triangulation of that polygon is given by the formula: \(t = n - 2\). This means that any polygon with \(n\) vertices can be divided into \(n-2\) triangles.

Base Case for Induction

Start with the base case where \(n = 3\), which is a triangle. A triangle already has 3 vertices and itself is a single triangle, so \(t = 3 - 2 = 1\). This matches the formula \(t = n - 2\).

Induction Hypothesis

Assume that the relationship holds for any polygon with \(k\) vertices, meaning it can be divided into \(k - 2\) triangles. This is our induction hypothesis.

Inductive Step

Prove the formula for a polygon with \(k + 1\) vertices. Assume you have a polygon with \(k + 1\) vertices. Remove one triangle by drawing a diagonal that connects non-adjacent vertices, which divides the polygon into a triangle and a smaller polygon with \(k\) vertices. By the induction hypothesis, this smaller polygon can be divided into \(k - 2\) triangles. Add the one triangle that was removed back, and you have \((k - 2) + 1 = k - 1 = (k + 1) - 2\) triangles in total.

Conclusion

Since both the base case and the inductive step have been verified, by the principle of mathematical induction, the relationship \(t = n - 2\) holds true for any polygon with \(n\) vertices.

Key concepts

Polygon Triangulation

Polygon triangulation is a process where a polygon with multiple vertices is divided into smaller, non-overlapping triangles. This concept is essential in computational geometry and various applications like computer graphics and geographical mapping. The key formula to understand here is that a polygon with \(n\) vertices can always be divided into \(n - 2\) triangles. This applies whether the polygon is convex or concave, as long as it is simple, meaning its sides do not intersect each other except at vertices.

• Triangulation is crucial because it simplifies complex problems into more manageable parts — triangles.
• This division allows for easier computation and analysis of various properties of the polygon.
• Uses are found in rendering 3D objects and modeling surfaces in digital spaces.

Induction Hypothesis

The induction hypothesis is a critical part of proving mathematical statements using induction. It involves assuming that a given statement is true for a certain integer \(k\), and then showing that it must also be true for \(k+1\). This step allows us to extend the truth from a specific instance to a broader range of cases.In our particular problem of polygon triangulation, the hypothesis assumes that for any polygon with \(k\) vertices, the relationship \(t = k - 2\) (where \(t\) is the number of triangles) holds. This assumption is then utilized to prove that the relation is also valid for a polygon with \(k + 1\) vertices.

• The induction hypothesis is a stepping stone — it provides the foundation needed to justify the next step.
• It is assumed to be true to help argue the correctness of the formula for additional vertices.
• This hypothesis links specific known cases to a solution that applies to all cases.

Base Case

The base case serves as the foundation of any mathematical induction proof. It is the simplest instance of the problem being considered, often where the smallest number of elements is involved. Establishing the base case is critical because it provides the initial certainty from which we can inductively progress.In our problem of proving the triangulation formula, the base case is evident when \( n = 3 \). A polygon with three vertices is simply a triangle that indeed contains only one triangle, satisfying our formula \( t = n - 2\). This straightforward verification acts as the starting point for extended reasoning in the inductive step.

• The base case shows that the formula works at the simplest level.
• It's crucial that the base case holds true because it is the anchor for the induction process.
• Without a valid base case, the chain of reasoning in induction falls apart.

Inductive Step

The inductive step is where you demonstrate that the truth of the statement for one case (the induction hypothesis) ensures its truth for the next case. This step relies heavily on the assumption made during the induction hypothesis.To prove the formula \(t = n - 2\) for polygons, we use the inductive step by taking a polygon with \(k+1\) vertices. By removing a triangle from it, you create a smaller polygon with \(k\) vertices. Our hypothesis tells us this smaller polygon can be divided into \(k - 2\) triangles. Adding back the removed triangle shows the larger polygon has \((k - 2) + 1 = k - 1 = (k + 1) - 2\) triangles.

• This step is where the true power of induction shows — it builds on the established base case and hypothesis to prove more complex situations.
• Failure to properly demonstrate this step means the proof doesn't universally hold.
• This ensures the calculated relationship holds for all reasonable values.