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Chapter 6: Problem 7

7\. If \(\mathrm{P}\) is the set of parallelograms, and \(\mathrm{T}\) is the set of trapeziums, then \(\mathrm{P} \cap \mathrm{T}\) is (1) P. (2) \(\mathrm{T}\). (3) \phi. (4) None of these

Short Answer

Expert verified
Answer: The intersection between set P and set T is an empty set, denoted as \phi.

Step by step solution

01

Understanding Parallelograms and Trapeziums

A parallelogram is a quadrilateral with opposite sides parallel. A trapezium is a quadrilateral with only one pair of parallel sides. As we can see, both shapes are quadrilaterals, but their properties concerning parallel sides are different.
02

Comparing Parallelograms and Trapeziums

We can see that a parallelogram has two pairs of parallel sides, while a trapezium has only one pair of parallel sides. Therefore, for a shape to be both a parallelogram and a trapezium, it should satisfy the conditions for both sets.
03

Finding the Intersection

Since parallelograms have two pairs of parallel sides and trapeziums have one pair of parallel sides, there is no shape that can be both a parallelogram and a trapezium. As a result, the intersection between set P and set T is an empty set, denoted as \phi. Therefore, the correct answer is: (3) \phi.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallelogram Properties
A parallelogram is a type of four-sided shape known as a quadrilateral. One of its standout features is that both pairs of its opposite sides are parallel. This means they run alongside each other and will never meet, no matter how far they are extended.

Some important properties of parallelograms include:
  • Opposite sides are equal in length.
  • Opposite angles are equal, ensuring the shape maintains balance.
  • The diagonals bisect each other, or in simpler terms, each diagonal cuts the other into two equal parts.
  • The adjacent angles sum up to 180 degrees, making it a useful figure in geometry.
These properties make the parallelogram unique and distinct within the family of quadrilaterals. Understanding these traits helps in identifying parallelograms in various geometric problems and real-life applications.
Trapezium Properties
A trapezium, also known as a trapezoid in some regions, is another type of quadrilateral. What sets it apart is having only one pair of parallel sides. These parallel sides are called the "bases" of the trapezium.

Here are some specific properties that define a trapezium:
  • One pair of opposite sides is parallel, making them the bases.
  • It can have right angles, but it's not a requirement鈥攎any trapeziums have no right angles.
  • The non-parallel sides are known as "legs," and they can be of differing lengths.
  • The height of a trapezium is measured as the perpendicular distance between the two bases.
Trapeziums are versatile shapes encountered in various problem-solving scenarios, especially those involving parallel lines.
Intersection of Sets
The concept of the intersection of sets in mathematics refers to the common elements shared by two or more sets. When finding the intersection of two sets, we're looking for elements that appear in both sets simultaneously.

Using the example of parallelograms and trapeziums, let's explore the intersection:
  • A parallelogram needs two pairs of parallel sides, making it radically different in structure compared to a trapezium.
  • A trapezium has only one pair of parallel sides, so it cannot simultaneously meet the definition of a parallelogram.
  • Therefore, no quadrilateral can satisfy both definitions at the same time.
This means that the intersection of these two sets is the empty set, often written \(\phi\). For students new to set theory, recognizing and understanding intersections can be vital in navigating more complex mathematical scenarios.

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