The Vertices Of A Quadrilateral On The Coordinate Plane Are \[$(2,4),(-4,-2),(-2,4),\$\] And \[$(4,-2)\$\]. What Type Of Quadrilateral Has These Vertices?A. Rectangle B. Trapezoid C. Square D. Parallelogram

Understanding the Problem

To determine the type of quadrilateral formed by the given vertices, we need to analyze the properties of each type of quadrilateral and compare them with the given coordinates.

Properties of Different Types of Quadrilaterals

Rectangle

A rectangle is a quadrilateral with four right angles and opposite sides of equal length. The diagonals of a rectangle bisect each other.

Trapezoid

A trapezoid is a quadrilateral with one pair of parallel sides. The diagonals of a trapezoid are not necessarily perpendicular.

Square

A square is a quadrilateral with four right angles and all sides of equal length. The diagonals of a square bisect each other at right angles.

Parallelogram

A parallelogram is a quadrilateral with opposite sides of equal length and parallel. The diagonals of a parallelogram bisect each other.

Analyzing the Given Vertices

The given vertices are {(2,4),(-4,-2),(-2,4),$}$ and {(4,-2)$}$. To determine the type of quadrilateral, we need to find the lengths of the sides and the diagonals.

Finding the Lengths of the Sides

To find the lengths of the sides, we can use the distance formula:

${d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

Let's find the lengths of the sides:

• Side 1: {(2,4)$] to [(-4,-2)\$} ${d_1 = \sqrt{(-4 - 2)^2 + (-2 - 4)^2} = \sqrt{36 + 36} = \sqrt{72}}$
• Side 2: {(-4,-2)$] to [(-2,4)\$} ${d_2 = \sqrt{(-2 + 4)^2 + (4 + 2)^2} = \sqrt{4 + 36} = \sqrt{40}}$
• Side 3: {(-2,4)$] to [(4,-2)\$} ${d_3 = \sqrt{(4 + 2)^2 + (-2 - 4)^2} = \sqrt{36 + 36} = \sqrt{72}}$
• Side 4: {(4,-2)$] to [(2,4)\$} ${d_4 = \sqrt{(2 - 4)^2 + (4 + 2)^2} = \sqrt{4 + 36} = \sqrt{40}}$

Finding the Lengths of the Diagonals

To find the lengths of the diagonals, we can use the distance formula:

${d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

Let's find the lengths of the diagonals:

• Diagonal 1: {(2,4)$] to [(-2,4)\$} ${d_5 = \sqrt{(-2 - 2)^2 + (4 - 4)^2} = \sqrt{16 + 0} = \sqrt{16} = 4}$
• Diagonal 2: {(-4,-2)$] to [(4,-2)\$} ${d_6 = \sqrt{(4 + 4)^2 + (-2 + 2)^2} = \sqrt{64 + 0} = \sqrt{64} = 8}$

Determining the Type of Quadrilateral

Based on the lengths of the sides and the diagonals, we can determine the type of quadrilateral.

• The sides {d_1 = \sqrt{72}$] and [$d_3 = \sqrt{72}\$] are equal, and the sides \[$d_2 = \sqrt{40}$] and [$d_4 = \sqrt{40}$] are equal.
• The diagonals [d_5 = 4}$ and {d_6 = 8}$ are not equal.

Based on these properties, we can conclude that the given vertices form a rectangle.

The final answer is A. Rectangle.

Introduction

Quadrilaterals are a fundamental concept in geometry, and understanding their properties and types is essential for solving problems and making connections to real-world applications. In this article, we will explore the properties and types of quadrilaterals, and provide answers to frequently asked questions.

Q: What is a quadrilateral?

A: A quadrilateral is a polygon with four sides and four vertices. It is a two-dimensional shape that can be classified into different types based on its properties.

Q: What are the properties of a quadrilateral?

A: The properties of a quadrilateral include:

• Four sides
• Four vertices
• Two pairs of opposite sides
• Two pairs of opposite angles
• Diagonals that bisect each other (in some cases)

Q: What are the different types of quadrilaterals?

A: The different types of quadrilaterals include:

• Rectangle
• Square
• Trapezoid
• Parallelogram
• Rhombus
• Kite

Q: What is a rectangle?

A: A rectangle is a quadrilateral with four right angles and opposite sides of equal length. The diagonals of a rectangle bisect each other.

Q: What is a square?

A: A square is a quadrilateral with four right angles and all sides of equal length. The diagonals of a square bisect each other at right angles.

Q: What is a trapezoid?

A: A trapezoid is a quadrilateral with one pair of parallel sides. The diagonals of a trapezoid are not necessarily perpendicular.

Q: What is a parallelogram?

A: A parallelogram is a quadrilateral with opposite sides of equal length and parallel. The diagonals of a parallelogram bisect each other.

Q: What is a rhombus?

A: A rhombus is a quadrilateral with all sides of equal length. The diagonals of a rhombus bisect each other at right angles.

Q: What is a kite?

A: A kite is a quadrilateral with two pairs of adjacent sides of equal length. The diagonals of a kite are perpendicular.

Q: How do I determine the type of quadrilateral?

A: To determine the type of quadrilateral, you can use the following steps:

1. Check if the quadrilateral has four right angles. If it does, it is a rectangle or a square.
2. Check if the quadrilateral has opposite sides of equal length. If it does, it is a parallelogram or a rhombus.
3. Check if the quadrilateral has one pair of parallel sides. If it does, it is a trapezoid.
4. Check if the quadrilateral has two pairs of adjacent sides of equal length. If it does, it is a kite.

Q: What are the applications of quadrilaterals?

A: Quadrilaterals have numerous applications in real-world scenarios, including:

• Architecture: Quadrilaterals are used in the design of buildings, bridges, and other structures.
• Engineering: Quadrilaterals are used in the design of machines, mechanisms, and other devices.
• Art: Quadrilaterals are used in the creation of geometric patterns and designs.
• Science: Quadrilaterals are used in the study of geometry, trigonometry, and other mathematical concepts.

Conclusion

Quadrilaterals are a fundamental concept in geometry, and understanding their properties and types is essential for solving problems and making connections to real-world applications. By following the steps outlined in this article, you can determine the type of quadrilateral and apply its properties to various scenarios.