Question

Is the shape is a parallelogram? (1) The diagonals bisect each other (2) It has four 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

A. 1 alone, not 2 alone

Step-by-step solution

- Analyze Condition 1

Condition (1) states that the diagonals bisect each other. A property of a parallelogram is that its diagonals bisect each other. Hence, if the diagonals bisect each other, this confirms that the shape is a parallelogram.

- Analyze Condition 2

Condition (2) states that the shape has four sides. This condition is not sufficient to determine if a shape is a parallelogram, as many shapes, such as rectangles, squares, and trapezoids, have four sides.

- Combine Conditions

Although Condition (2) is true for all parallelograms, it alone is not enough to identify the shape specifically as a parallelogram. Condition (1) by itself is sufficient since it directly confirms the property of the diagonals that is unique to parallelograms.

Conclusion - Decide Between Answer Choices

Based on the analysis, Condition (1) alone is sufficient to determine that the shape is a parallelogram. Therefore, the correct answer is A. 1 alone, not 2 alone.

Key concepts

parallelogram properties

A parallelogram is a special type of quadrilateral that has specific properties which distinguish it from other quadrilaterals. To recognize a parallelogram, you should be familiar with the following properties:

• Opposite sides are parallel and of equal length.
• Opposite angles are equal.
• The diagonals bisect each other.
• Adjacent angles add up to 180 degrees.

These characteristics help in identifying a parallelogram among various quadrilaterals. Understanding these properties is essential, especially when approached with problems like in the given GMAT exercise.

GMAT problem solving

Solving GMAT geometry problems efficiently requires a solid understanding of geometric properties and logical reasoning. For the given problem, we need to determine if the shape is a parallelogram based on two conditions.
Condition (1): The diagonals bisect each other. This is a definitive property of parallelograms. By knowing this, we can confirm the shape is indeed a parallelogram.
Condition (2): The shape has four sides. Many shapes in geometry have four sides, such as rectangles, squares, and trapezoids. Hence, this condition alone does not provide enough information to confirm the shape as a parallelogram.
Combining the logic, understanding the properties, and knowing the specific characteristics are key to solving GMAT geometry problems.

diagonal bisection in parallelograms

One of the unique properties of a parallelogram includes the behavior of its diagonals. In a parallelogram, the diagonals bisect each other. This means each diagonal cuts the other into two equal halves.
Mathematically, if you have a parallelogram ABCD with diagonals AC and BD, these diagonals will intersect at a point E. Here, AE = EC and BE = ED.
This property is crucial in solving geometry problems that involve parallelograms, as it can be used to confirm the identity of the shape even when other information is ambiguous or missing.

shape identification

Identifying shapes accurately is fundamental in geometry problems. When provided with specific characteristics, as seen in the GMAT problem, you need to apply known properties to determine the shape.
For example, if the diagonals of the shape bisect each other, you can confidently conclude that the shape is a parallelogram.
Conversely, knowing only that a shape has four sides does not help much, as it could refer to various shapes like rectangles, squares, trapezoids, etc.
Hence, precise properties and distinctive traits are necessary for accurate shape identification, making the use of all given information and geometric characteristics critical.