Question

Can a polygon be divided into three triangles? (1) It is a pentagon (2) It has five sides A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

D. 1 alone or 2 alone

Step-by-step solution

Understanding the Problem

The problem is asking whether a polygon can be divided into three triangles based on given information. We need to analyze two statements: (1) the polygon is a pentagon, and (2) the polygon has five sides.

Analyze Statement (1)

A pentagon is a polygon with 5 sides. By definition, a pentagon can be divided into three triangles by drawing diagonals from one vertex. This means statement (1) alone is sufficient to conclude that the polygon can be divided into three triangles.

Analyze Statement (2)

Statement (2) also indicates the polygon has five sides, which is the definition of a pentagon. Thus, statement (2) alone is also sufficient to conclude that the polygon can be divided into three triangles.

Combine Statements

Since both statements essentially provide the same information, combining them does not provide any additional insight. Each statement alone is sufficient to conclude that the polygon can be divided into three triangles.

Conclusion

Based on the analysis, either statement (1) alone or statement (2) alone is sufficient to determine that the polygon can be divided into three triangles.

Key concepts

polygon properties

A polygon is a flat, closed shape with straight sides. The number of sides determines the type of polygon.
Some common polygons include triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), and hexagons (6 sides).

Each polygon has unique properties, such as:

• The sum of its internal angles. For a polygon with n sides, this is given by the formula: \((n-2) \times 180°\).
• The number of diagonals, which can be calculated using the formula: \(\frac{n(n-3)}{2}\).

For instance, a pentagon has five sides. It can be divided into three triangles by drawing two diagonals from a single vertex. This is a key property that can help solve various geometry problems, including those on the GMAT.

triangle division

Dividing a polygon into triangles helps to simplify complex problems. By breaking down a polygon into smaller, manageable components, we can more easily calculate areas and angles.

To divide a polygon into triangles:

• Start at one vertex and draw diagonals to non-adjacent vertices.
• For a polygon with n sides, you will be able to draw exactly \(n-2\) triangles.
  For example, in a pentagon (5 sides), drawing diagonals from one vertex will yield three triangles.

This technique is beneficial in analytical reasoning, particularly in standardized tests like the GMAT.

analytical reasoning

Analytical reasoning involves breaking down information systematically and logically. It is crucial for solving problems efficiently.

Here's how to apply it:

• Understand the problem: Identify what is being asked.
• Examine given statements: Determine their relevance to the problem.
• Use known properties: Apply mathematical principles like those of polygons and triangles.
• Synthesize information: Combine all insights to draw a final conclusion.

In this exercise, we first identified that the problem involves polygon properties, then we connected this to the specific attributes of pentagons and triangles.

mathematical logic

Mathematical logic forms the foundation of problem-solving. It includes principles and methodologies that help in deriving conclusions from given premises.

Key steps in using mathematical logic:

• Identify known facts: Use given information, like the polygon having five sides.
• Apply definition-based reasoning: Recognize that a five-sided polygon is a pentagon.
• Draw logical conclusions: If a pentagon, it can be divided into three triangles based on its properties.

In this GMAT problem, by logically dissecting the statements, we determined that either statement alone is sufficient to conclude that the polygon can be divided into three triangles. This method of verifying both independent and combined sufficiency is essential for GMAT success.