A **rectangle** is a fundamental geometric shape with several important properties and applications. Here's a comprehensive overview: ### **Definition** A **rectangle** is a quadrilateral (a four-sided polygon) where every interior angle is a right angle (90 degrees). This definition implies several key characteristics: - **Opposite Sides are Equal and Parallel:** In a rectangle, opposite sides have equal lengths and run parallel to each other. - **Diagonals are Equal:** The two diagonals of a rectangle are congruent (they have the same length) and bisect each other. ### **Properties** 1. **Angles:** All four angles in a rectangle are right angles (90°). 2. **Sides:** - **Opposite Sides:** Equal in length (`length = length`, `width = width`). - **Adjacent Sides:** Perpendicular to each other. 3. **Diagonals:** - **Equal Length:** Both diagonals have the same length. - **Bisect Each Other:** They cut each other exactly in half. 4. **Symmetry:** A rectangle has two lines of symmetry (through the midpoints of opposite sides) and rotational symmetry of order 2. ### **Formulas** 1. **Perimeter (P):** \[ P = 2 \times (\text{Length} + \text{Width}) \quad \text{or} \quad P = 2l + 2w \] 2. **Area (A):** \[ A = \text{Length} \times \text{Width} \quad \text{or} \quad A = l \times w \] 3. **Diagonal (d):** Using the Pythagorean theorem: \[ d = \sqrt{l^2 + w^2} \] Where $$ l $$ is the length and $$ w $$ is the width of the rectangle. ### **Special Cases** - **Square:** A square is a special type of rectangle where all four sides are of equal length. This means a square inherits all properties of a rectangle, with the additional property of equal sides. ### **Applications** Rectangles are ubiquitous in both theoretical mathematics and practical everyday scenarios: - **Architecture and Engineering:** Used in designing rooms, buildings, and various structures. - **Design and Art:** Common in graphic designs, digital screens, and canvases. - **Mathematics:** Fundamental in studying geometrical principles, area and volume calculations, and coordinate geometry. - **Everyday Objects:** Books, screens (phones, TVs, monitors), tables, doors, and many other objects are rectangular. ### **Examples** 1. **Calculating Area and Perimeter:** - If a rectangle has a length of 8 units and a width of 3 units: - **Area:** $$ 8 \times 3 = 24 $$ square units. - **Perimeter:** $$ 2(8 + 3) = 22 $$ units. 2. **Finding the Diagonal:** - For the same rectangle: - $$ d = \sqrt{8^2 + 3^2} = \sqrt{64 + 9} = \sqrt{73} \approx 8.54 $$ units. ### **Visual Representation** ``` +----------------+ | | | | Width (w) | | +----------------+ Length (l) ``` ### **Coordinate Geometry Perspective** In a coordinate plane, a rectangle can be defined by four vertices where opposite sides are parallel to the axes or have the same slope. For a rectangle with vertices at $$(x_1, y_1)$$, $$(x_2, y_1)$$, $$(x_2, y_2)$$, and $$(x_1, y_2)$$, the side lengths and area can be easily computed using the coordinate distances. ### **Conclusion** Understanding rectangles lays the foundation for more advanced geometric concepts and has practical implications in various fields. Their simple yet robust properties make them a key shape in both academic studies and real-world applications. A **rectangle** is a four-sided shape with all right angles, opposite sides equal and parallel, and diagonals that are equal in length and bisect each other.

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2. rectangle

Ask by Ross Warner. in the United States

Feb 04,2025

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A **rectangle** is a four-sided shape with all right angles, opposite sides equal and parallel, and diagonals that are equal in length and bisect each other.

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Did you know that rectangles have a fascinating historical background? The concept of a rectangle dates back to ancient civilizations, where Egyptians used a simple four-sided figure to lay out their fields and structures. The word "rectangle" itself comes from the Latin "rectangulus," meaning "right angle." This geometric shape has been essential in design, architecture, and engineering for centuries! In the real world, rectangles are everywhere! From the screens of your smartphone to the shape of a classic book cover, rectangles dominate both functional and aesthetic aspects of our lives. Their properties make them ideal for creating efficient spaces in architecture and art, illustrating just how versatile and practical this shape is in everyday applications!

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