Rectangle

8 Key excerpts on "Rectangle"

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  Mathematical Literacy NQF2 SB

  TVET FIRST

  • K van Niekerk O Roberts(Author)
  • 2017(Publication Date)
  • Macmillan

    (Publisher)

  105 Module 6 Two-dimensional shapes and three-dimensional objects A small angle with less turn and a larger angle with more turn. Right angles The sides of squares and Rectangles form right angles at the corners. A right angle measures 90°. The square symbols on the diagrams show the right angles. P Q R S T U V The symbols at the corners of Rectangle STUV show that the angles are right angles. Triangle PQR is a right-angled triangle, because it has a right angle at Q. If two line segments are at right angles to each other, they are perpendicular to each other. So PQ is perpendicular to QR in the diagram given. Parallel lines Parallel lines are always exactly the same distance apart and they never touch each other. We show that a pair of lines is parallel by drawing arrowheads or double arrowheads on the lines. In Rectangle ABCD, AB is parallel to CD and AD is parallel to BC. Describing shapes in terms of their properties Rectangles A Rectangle is a 2-D shape with four straight sides. A Rectangle always has both pairs of opposite sides equal in length, and all four angles are right angles. Look at how these properties are marked on this diagram alongside: A B C D Right angle: An angle that measures exactly 90°, such as the corner of a square. New words J K L M 106 Module 6 Topic 2: Space, shape and orientation Squares A square is a 2-D shape with four straight sides. Squares are shapes with four straight sides, with all sides equal in length and all the angles are right angles. A straight line from one vertex of a shape to the opposite vertex is called a diagonal . The two diagonals of a square or Rectangle are equal in length. The diagonals of Rectangle ABCD are AC and BD. The diagonals divide the Rectangle into four large triangles. Can you use letters ABCD to name the four triangles? Triangles A closed shape with three straight sides is a triangle. The sides can be different lengths and the angles can be different sizes.

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  Helping Children Learn Mathematics

  • Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith, Anna Rogers, Audrey Cooke, Bronwyn Ewing, Kylie Robson(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Thus, a square, a rhombus and a Rectangle are all special types of parallelogram. Discuss these questions with a peer. A rhombus is a parallelogram with all sides congruent (equal in length). Does that mean that a square is a rhombus? Explain why or why not. A Rectangle is a special parallelogram with all right angles. Does that mean a square is a Rectangle? Explain why or why not. Through the centuries, definitions of shapes have changed. For example, once there were oblongs and squares, but the shape name Rectangle is now used for a quadrilateral with four right angles, which is not a square. It is best not to use the term oblong in geometry, as it can be confused with other real-world concepts and applications which are mathematically different. Helping children learn spatial relationships and classifications takes time. Children first begin by verbalising properties of shapes. To build a deeper understanding, they need to visualise shapes and their properties according to Seah (2015). By engaging in paper-folding activities (of triangles, squares and polygons), children will be able to view two-dimensional shapes from different orientations. In particular, the properties of quadrilaterals according to their length of side and size of angles will become much clearer. For example, a square can be described as: • a closed, four-sided figure (property 1) • having opposite sides parallel (property 2) Pdf_Folio:566 566 Helping children learn mathematics • having all right angles (property 3) • having all sides congruent, equal in length (property 4). Properties 1 and 2 make it a parallelogram; properties 1, 2 and 3 make it a Rectangle; properties 1, 2 and 4 make it a rhombus; properties 1, 2, 3 and 4 make it a square. Additionally, a square has the rare property that diagonals bisect each other at 90 degrees (also found in a kite and rhombus). In the classroom 16.12 helps children with this idea.

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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  CHAPTER OUTLINE 4.1 Properties of a Parallelogram 4.2 The Parallelogram and Kite 4.3 The Rectangle, Square, and Rhombus 4.4 The Trapezoid ■ PERSPECTIVE ON HISTORY: Sketch of Thales ■ PERSPECTIVE ON APPLICATIONS: Square Numbers as Sums ■ SUMMARY Chapter 4 © Richard A. Cooke/CORBIS. 169 Quadrilaterals Comforting! Designed by architect Frank Lloyd Wright (1867–1959), this private home is nestled among the trees in the Bear Run Nature Preserve of southwestern Pennsylvania. Known as Fallingwater, this house was constructed in the 1930s. The geometric figure that dominates the homes designed by Wright is the quadrilat-eral. In this chapter, we consider numerous types of quadrilaterals— among them the parallelogram, the rhombus, and the trapezoid. Also, the language and properties for each type of quadrilateral are developed. Each type of quadrilateral has its own properties and applications. Many of these real-world applications can be found in the examples and exercises of Chapter 4. Additional video explanations of concepts, sample problems, and applications are available on DVD. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 170 CHAPTER 4 ■ QUADRILATERALS Unless otherwise noted, all content on this page is © Cengage Learning. A quadrilateral is a polygon that has exactly four sides. Unless otherwise stated, the term quadrilateral refers to a plane figure such as ABCD in Figure 4.1(a), in which the line seg-ment sides lie within a single plane.

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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  uni2220NMQ _ uni2220PQM 7. uni25B3NMQ _ uni25B3PQM 8. MP _ QN 1. Given 2. By definition, a Rectangle is a uni25B1 with a right angle 3. Opposite sides of a uni25B1 are _ 4. Identity 5. By Corollary 4.3.1, the four uni2220s of a Rectangle are right uni2220s 6. All right uni2220s are _ 7. SAS 8. CPCTC A B D C Square ABCD Figure 4.22 200 CHAPTER 4 ■ QUADRILATERALS Because a square is a type of Rectangle, it has four right angles and its diagonals are congruent. Because a square is also a parallelogram, its opposite sides are parallel. For any square, we can show that the diagonals are perpendicular; see Exercise 34. In Chapter 8, we measure area in “square units.” For the calculation of area, we count the number of congruent squares (square units) that fit inside a geometric region. THE RHOMBUS The next type of quadrilateral we consider is the rhombus. The plural of the word rhombus is rhombi (pronounced rho ˘m-bi ¯ ). While this definition requires only two congruent sides, the following corollary deter- mines that a square actually has four congruent sides. Corollary 4.3.3 All sides of a square are congruent. Corollary 4.3.4 All sides of a rhombus are congruent. Corollary 4.3.5 The diagonals of a rhombus are perpendicular. SSG EXS. 5–7 A rhombus is a parallelogram with two congruent adjacent sides. (See Figure 4.23.) DEFINITION In Figure 4.23, the adjacent sides AB and AD of rhombus ABCD are marked congru- ent. Because a rhombus is a type of parallelogram, it is also necessary that AB _ DC and AD _ BC . Thus, we have Corollary 4.3.4. We will use Corollary 4.3.4 in the proof of the following corollary. To visualize Corollary 4.3.5, see Figure 4.24(a). Discover What type of quadrilateral is deter- mined when an isosceles triangle is reflected across its base? ANSWER Rhombus EXAMPLE 2 Study the picture proof of Corollary 4.3.5. In the proof, pairs of triangles are congruent by the reason SSS.

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  Geometry

  A Self-Teaching Guide

  • Steve Slavin, Ginny Crisonino(Authors)
  • 2004(Publication Date)
  • Wiley

    (Publisher)

  Squares, Rectangles, trapezoids, and parallelograms are all quadrilaterals. And a quadrilateral, as you may recall from chapter 1, is a four-sided polygon. Squares As you might well know, a square contains four right angles (angles of 90°) and has four sides of equal length. s s s s 126 GEOMETRY Perimeter of a square formula: P = 4s, where s stands for the length of a side. Here’s a nice easy one: Find the perimeter of the following square. P = 4s = 4(2) = 8. The perimeter is 8 feet. Now here’s a challenge. Find the area, A, of this same square. Area of a square formula: A = s 2 A = s 2 = 2 2 = 4. The area is 4 square feet. Rectangles A Rectangle contains four right angles (90°), a pair of equal lengths, and a pair of equal widths. Perimeter of a Rectangle formula: P = 2l + 2w Using this formula, find the perimeter of the following Rectangle. P = 2l + 2w = 2(4) + 2(6) = 8 + 12 = 20. The perimeter is 20 inches. w = 4 inches = 6 inches w w s = 2 feet Perimeter and Area of Two-dimensional Polygons 127 Area of a Rectangle formula: A = lw Go ahead and find the area of this Rectangle. A = lw = 6(4) = 24. The area is 24 square inches. Example 1: Find the area of a Rectangle if the length is 15 and the perimeter is 50. Solution: To find the area, first we have to find the width. We’ll use the formula for perime- ter to find the width. P = 2l + 2w Substitute 15 for l and 50 for P. 50 = 2(15) + 2w 50 = 30 + 2w Subtract 30 from both sides of the equation. 20 = 2w Divide both sides of the equation by 2. 10 = w The width is 10. Now that we know the width, we can substitute its value into the formula for area. A = lw A = 15(10) = 150 Example 2: Find the perimeter of a Rectangle if its width is 9 inches and its area is 117 square inches. Solution: To find the perimeter, we have to find the length of the Rectangle. We know the area, so we’ll start by using the area formula to find the length.

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  Years 9 - 10 Maths For Students

  • (Author)
  • 2015(Publication Date)
  • For Dummies

    (Publisher)

  In the previous section, the perimeter formulas deal with linear measure. Linear measure is just one dimension. It’s from one place to another — there’s no breadth to it. You measure it with a ruler or tape measure in one direction. Square measurements are used to measure area. Area takes two measures — one along a side and a second perpendicular (90 degrees) to that side. Laying out Rectangles and squares Rectangles and squares have basically the same area formulas because they both have square corners and the equal lengths on opposite sides. The general procedure here is just to multiply the measure of the length times the measure of the width. The product of two sides that are next to one another is the area. Finding the area of a Rectangle or square Most rooms in homes, schools and offices are rectangular in shape. Desks and tables and rugs are usually rectangular, also. This makes it easy to fit furniture and other objects in the room. The area of a Rectangle is its length times its width, and the area of a square is the square of the measure of any side: Rectangle: A = l × w Square: A = s 2 Say a garden 35 metres long by 15 metres wide needs some fertiliser. If a bag of fertiliser covers 6 square metres, how much fertiliser do you need? First determine how many square metres the garden is. area of garden = l × w = 35 × 15 = 525 square metres Divide the 525 square metres by 6 square metres: 525 ÷ 6 = 87.5 square metres You can buy 88 bags and have some left over, or buy 87 bags and skimp a little in some places. 322 Part IV: Applying Algebra and Understanding Geometry Tuning in triangles Finding the area of a triangle can be a bit of a challenge. Basically, a triangle’s area is half that of an imaginary Rectangle that the triangle fits into. However, it isn’t always easy or necessary to find the length and width of this hypothetical Rectangle — you just need a measurement or two from the triangle.

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  Dr. Math Introduces Geometry

  Learning Geometry is Easy! Just ask Dr. Math!

  • (Author)
  • 2004(Publication Date)
  • Jossey-Bass

    (Publisher)

  Keep your quadrilateral definitions handy, and check to see if the diagram makes sense to you. Here are some of the things it should tell you. Some quadrilaterals are kites, some are trapezoids, and some are scalene quadrilaterals. Some trapezoids are parallelo- grams, some are isosceles, and some are neither. Parallelograms that are also isosceles trapezoids are Rectangles; those that are both isosceles trapezoids and rhombuses are squares. Not only are all Rectangles parallelograms, but all of the proper- ties of parallelograms are true for Rectangles. Two properties of par- allelograms are that the opposite sides are parallel and the diagonals bisect each other. Since Rectangles and rhombuses are parallelograms, then they also have opposite sides that are paral- lel and diagonals that bisect each other. Note that I am using the definition of a trapezoid that says that at least one pair of the sides must be parallel. If we have two sides that are parallel, then it’s also a parallelogram. Some math books use a different definition in which exactly one pair of sides is parallel. —Dr. Math, The Math Forum 46 Dr. Math Introduces Geometry R esources on the Web Learn more about two-dimensional geometric figures at these sites: Math Forum: Ask Dr. Math: Point and Line mathforum.org/library/drmath/view/55297.html A point has no dimension (I’m assuming), and a line, which has dimension, is a bunch of points strung together. How does some- thing without dimension create something with dimension? Math Forum: Problems of the Week: Middle School: Back Yard Trees mathforum.org/midpow/solutions/solution.ehtml?puzzle=35 How many different quadrilaterals can be formed by joining any four of the nine trees in my backyard? Math Forum: Problems of the Week: Middle School: Picture-Perfect Geometry mathforum.org/midpow/solutions/solution.ehtml?puzzle=97 Graph four points and name the figure that you have drawn.

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  Philosophy of Mathematics and Deductive Structure in Euclid's Elements

  • Ian Mueller(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  4

  Proportion and the Geometry of Plane Rectilineal Figures

  Among the elementary facts of traditional geometry are formulas for computing the areas and volumes of figures on the basis of certain lengths. For example, The area of a parallelogram is the product of its base and its height;

  The area of a triangle is of the product of its base and its height.

  Knowledge of such formulas can be traced with reasonable probability to a period long before classical Greek civilization.1 The fact that such formulas are not to be found in the Elements or in other mathematical works of the third century is obviously not due to ignorance. To understand why these formulas are not found in the Elements one must first understand what conceptual apparatus would be required in a satisfactory proof of their validity.

  Clearly, the first thing needed is a number system with multiplication and a way of assigning numbers to lines and figures. But to assign a number to a line is not to make an abstract correlation; it is to say that the line has a certain length. Hence the formulas for areas presuppose a notion of signed quantities. A straight line is not 9 but 9 centimeters in length; a triangle is not 3 but 3 square meters in area, and so on. Obviously one needs as well an understanding of the relations between signed quantities; for example, one needs to know that the product of a length in meters and a length in meters is an area in square meters. Finally, it seems, one needs to understand notions like length, area, and volume abstractly. A figure isn’t an area; it has an area, an area which another figure might have as well.

  I have already indicated the general absence of abstract notions in Euclid’s geometry. The absence of the other concepts underlying the standard geometric formulas does not need to be documented. The assignment of numbers to geometric objects and, therefore, signed quantities are simply not to be found in the Elements.

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