Parallelograms

10 Key excerpts on "Parallelograms"

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  eBook - ePub

  A First Step to Mathematical Olympiad Problems

  • Derek Holton(Author)
  • 2009(Publication Date)
  • WSPC

    (Publisher)

  5.3. Rectangles and Parallelograms

  A rectangle is a four-gon (four-sided figure) all of whose angles are right angles and whose opposite sides are equal in length.

  We show a rectangle in Figure 5.7 . Obviously a square is a special type of rectangle.

  Exercises

  11. Do rectangles tessellate the plane? 12. Did Escher ever start one of his “tessellations” from rectangles? 13. Are rectangles self-replicating? 14. Is every shape that tessellates the plane a self-replicating shape? 15. Can you square a rectangle (i) with squares of equal size; (ii) with squares of unequal size; (iii) with squares which are all of different sizes;

  (iv) with m squares;

  (v) with one square each of side length 1,2,…, n? 16. Divide a rectangle of side lengths 6 and 9 into squares of side length one. How many squares are there?Generalise. 17. Can you rectangle a rectangle? That is, can you make up a rectangle from smaller rectangles? In what different ways can this be done? 18. Take any two rectangles and plonk them down anywhere in the plane. In how many different shapes will the two rectangles intersect?

  So now we get to Parallelograms. A parallelogram is a gram made of parallels. Take two pairs of parallel lines. The four-sided figure (“gram”) they make is a parallelogram (see Figure 5.8 ). So a parallelogram is a foursided figure with both pairs of opposite sides parallel.

  Figure 5.8.

  We represent the parallel property by the insertion of arrows. Because the top and bottom sides of the parallelogram in Figure 5.8 are parallel we put an arrow on each of them. Because the left and right sides of the parallelogram are parallel (but not parallel to the top and bottom sides) we put two arrows on each of them.

  In general the angles between adjacent sides of a parallelogram are not equal. However, when they are we get a rectangle or a square. Squares and rectangles are just special Parallelograms.

  Exercise

  19. Repeat Exercises 11-18 with the words “rectangle” and “square” replaced everywhere by “parallelogram”.

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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)

  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. When two sides of the quadrilateral are skew (not coplanar), as with MNPQ in Figure 4.1(b), that quadrilateral is said to be skew. Thus, MNPQ is a skew quadrilateral. In this textbook, we generally consider quadrilaterals whose sides are coplanar. Figure 4.1 C B A (a) D N M Q P (b) Quadrilateral Skew Quadrilateral Parallelogram Diagonals of a Parallelogram Altitudes of a Parallelogram KEY CONCEPTS Properties of a Parallelogram 4.1 V T R S (a) V T R S (b) D iscover From a standard sheet of construc-tion paper, cut out a parallelogram as shown. Then cut along one diagonal. How are the two triangles that are formed related? ANSWER 11 2 2 8 1 2 / They are congruent. Figure 4.2 A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. DEFINITION In Figure 4.2(a), if and , then RSTV is a parallelogram. We sometimes indicate parallel lines (and line segments) by showing arrows in the same direction. Thus, Figure 4.2(b) emphasizes the fact that RSTV is a parallelogram. Because the symbol for parallelogram is , the quadrilateral in Figure 4.2(b) is . The set is a subset of ; that is, . The Discover activity at the left hints at many of the theorems of this section. EXAMPLE 1 Give a formal proof of Theorem 4.1.1. P Q Q {quadrilaterals} P {Parallelograms} RSTV RV ST RS VT THEOREM 4.1.1 A diagonal of a parallelogram separates it into two congruent triangles. GIVEN: with diagonal (See Figure 4.3 on page 171.) PROVE: ACD CAB AC ABCD Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

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  eBook - ePub

  Geometry For Dummies

  • Mark Ryan(Author)
  • 2016(Publication Date)
  • For Dummies

    (Publisher)

  Five ways to prove that a quadrilateral is a parallelogram: There are five different ways of proving that a quadrilateral is a parallelogram. The first four are the converses of parallelogram properties (including the definition of a parallelogram). Make sure you remember the oddball fifth one — which isn’t the converse of a property — because it often comes in handy:

  • If both pairs of opposite sides of a quadrilateral are parallel, then it’s a parallelogram (reverse of the definition). Because this is the reverse of the definition, it’s technically a definition, not a theorem or postulate, but it works exactly like a theorem, so don’t sweat this distinction.
  • If both pairs of opposite sides of a quadrilateral are congruent, then it’s a parallelogram (converse of a property).
    To get a feel for why this proof method works, take two toothpicks and two pens or pencils of the same length and put them all together tip-to-tip; create a closed figure, with the toothpicks opposite each other. The only shape you can make is a parallelogram.
  • If both pairs of opposite angles of a quadrilateral are congruent, then it’s a parallelogram (converse of a property).
  • If the diagonals of a quadrilateral bisect each other, then it’s a parallelogram (converse of a property).
    Take, say, a pencil and a toothpick (or two pens or pencils of different lengths) and make them cross each other at their midpoints. No matter how you change the angle they make, their tips form a parallelogram.
  • If one pair of opposite sides of a quadrilateral are both parallel and congruent, then it’s a parallelogram (neither the reverse of the definition nor the converse of a property).
    Take two pens or pencils of the same length, holding one in each hand. If you keep them parallel, no matter how you move them around, you can see that their four ends form a parallelogram.

  The preceding list contains the converses of four of the five parallelogram properties. If you’re wondering why the converse of the fifth property (consecutive angles are supplementary)

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  Introduction to Mathematical Literacy

  • Alberto D. Yazon(Author)
  • 2019(Publication Date)
  • Society Publishing

    (Publisher)

  Figure 4.10: An example of polygon. Source: https://upload.wikimedia.org/wikipedia/commons/thumb/9/99/Hexa-gon.svg/2000px-Hexagon.svg.png. Perimeter and Area 83 4.5.8. The Parallelogram A parallelogram is another shape in the field of geometric shapes in which the opposite side of the shape is parallel. To have the potential to evaluate, if the sides are parallel or not, then the individual has to closely evaluate the shape. The main quality of a parallelogram is that parallel lines never cross or intersect each other, no matter how long any individual extend them. In this way, if an individual goes on increasing the lines with the help of eternity and they never intersect each other, then they can be stated as a parallelogram (Figure 4.11). Figure 4.11: An example of a parallelogram. Source: https://upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Romb. svg/1280px-Romb.svg.png. Never the less, if the lines touch or meet at any given point, then that shape cannot be well-thought-out a parallelogram. In the same way, a triangle cannot be well-thought-out a parallelogram in the meantime the lines opposite to a triangle meet at the point of the triangle. And in the meantime, the lines intersect; it cannot be stated as a parallelogram. 4.6. CIRCLE 4.6.1. Area of Circle In the field of geometry, the area surrounded with the help of a circle of having a radius r is πr 2 . Here the Greek letter π represents a constant, which is near about equal to the value of 3.14159, which is equal to the ratio of the circumference of any circle to its diameter. Introduction to Mathematical Literacy 84 One technique or method of deriving this formula, which invented with the help of the Archimedes, is consisting of viewing the circle as the limit of a sequence of regular polygons.

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  Primary Mathematics

  Integrating Theory with Practice

  • Penelope Serow, Rosemary Callingham, Tracey Muir(Authors)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  Parallelism Opposite sides are parallel. ....................................................................................................................................................................................................................................................................... Diagonals Diagonals bisect each other. Congruent figures figures that are exactly the same size and shape Transformation shifting or modifying a shape, including reflecting, enlarging, translating and rotating Bisect divide into two equal parts Dissect divide into two parts CHAPTER 4 EXPLORING GEOMETRY 85 of square, rectangle and rhombus into the class of Parallelograms. This is a difficult hurdle to overcome, and in doing so is a target outcome in the secondary setting. It is important for primary teachers to be aware of the importance of the early property explorations; otherwise, concepts such as symmetry and diagonal properties could possibly be overlooked. ACTIVITY 4.5 Design a similar template (as Table 4.2) for exploration of all the triangle and quadrilateral figures. What would be the differences between students’ descriptions of their findings at Level 1, 2 and 3 thinking? It is common for some students at Level 1 and Level 2 to describe the square as if it is the ‘king of the quads’. At Level 1, students may describe all other quadrilaterals as a morphed version of the square. This is evident in Classroom Snapshot 4.3. Classroom Snapshot 4.3 Peter’s teacher showed him all the quadrilaterals, one after another, on flashcards. He very quickly told the teacher the name of all the quadrilaterals. When the teacher asked Peter to describe the shapes he replied: The rectangle is a stretched out square, the parallelogram is a pushed over rectangle, the rhombus is a pushed over square, the kite is a stretched diamond and the trapezium is a rectangle with bits chopped off.

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  Geometry Transformed

  Euclidean Plane Geometry Based on Rigid Motions

  • James R. King(Author)
  • 2021(Publication Date)
  • American Mathematical Society

    (Publisher)

  Chapter 9 Area and Its Applications Areas of Triangles and Parallelograms Area, a familiar and useful topic in the real world, is not only an interesting topic in geometry; it also provides an additional tool for understanding geometrical relation-ships. While it is simple in some respects, it is not so simple in others. The simple part includes the familiar formulas for areas of rectangles, triangles, Parallelograms, and other basic polygons. The not-so-simple part is expressing a general definition of area and then proving it agrees with the simple formulas but also applies to shapes such as circles and other nonpolygonal shapes. To derive an area formula for circles, limits are involved. This is as far as we will be going in this book. 1 Area should have these properties: • Congruent figures have the same area. • For a square ? of side length 1, the area 𝒜(?) = 1 . • If the set ? 1 contains ? 2 , then 𝒜(? 1 ) ≥ 𝒜(? 2 ) . • If a set ? is the union of a finite number ? 1 , ? 2 , ... , ? 𝑘 of convex polygons and their interiors such that the polygon interiors do not intersect, then 𝒜(?) = 𝒜(? 1 ) + 𝒜(? 2 ) + ⋯ + 𝒜(? 𝑘 ). Squares and the Definition of Area. All our work with area will rely on the areas of squares. Consider how only one definition makes sense. A square of integer side 𝑚 can be divided into 𝑚 2 unit squares of side 1, so the area should be 𝑚 2 . For example, the square on the left of Figure 1 is 9, assuming the small squares have side 1. If the area of a square of side 1 is divided into squares of side length 1/𝑛 , then since there are 𝑛 2 such small squares, the area of the small square should be 1/𝑛 2 . The area of 1 If one investigates really complicated sets in the plane, such as fractals, it turns out that a set may have no area, or it may have more than one value competing to be area. 161

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  Linear Algebra and Geometry

  • Al Cuoco, Kevin Waterman, Bowen Kerins, Elena Kaczorowski(Authors)
  • 2019(Publication Date)
  • American Mathematical Society

    (Publisher)

  Since they are equivalent, you know they have the same length (magnitude), so the line segments ←− A convex quadrilateral is a parallelogram if one pair of opposite sides is both parallel and congruent, so there’s no need to show that −−→ BC is equivalent to −−→ AD . are congruent. They also have the same direction, so the line segments have the same slope and are thus parallel. For You to Do 5. Derman tried to show that he had a parallelogram if A = (2 , 1), B = (4 , 2), C = (6 , 3), and D = (10 , 5), and he ended up scratching his head. Help him figure out why these points do not form a parallelogram. Developing Habits of Mind Use vectors to describe geometric ideas. If points in R 2 and R 3 are viewed as vectors, the geometric description of addition and scalar multiplication is much easier. B B O If A and B are vectors in R 2 , A + B is the diagonal of the parallelogram whose sides are A and B . Multiplying A by c yields a vector whose length is the length of A multiplied by | c | ; cA has the same direction as A if c > 0; cA has the opposite direction of A if c < 0. 24 1.3 Vectors In Exercise 1 from Lesson 1.1, you plotted several points that were scalar ←− In Exercise 6, you’ll show that if P and Q are nonzero vectors in R 2 or R 3 , and that if O , P , and Q are collinear, then Q = cP for some real number c . In fact, the set of points collinear with the points O and P is the collection of multiples cP of P . multiples of a point A = (1 , 2). What you might have noticed was that all of the resulting points ended up on the same line: 10 8 6 4 2 0 0 2 4 6 8 -2 -2 -4 -6 -8 -10 -12 -14 -4 -6 A a. 2 A = (2,4) d. ( -1) A = ( -1, -2) e. ( -3) A = ( -3, -6) f. ( -6.5) A = ( -6.5, -13) b. 3 A = (3,6) c. 5 A = (5,10) The fact that scalar multiples of points are collinear hints at an algebraic way to characterize parallel vectors in R 2 . In the following figure, −−→ AB and Remember A point A corresponds to the vector −→ OA .

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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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  Dr. Math Presents More Geometry

  Learning Geometry is Easy! Just Ask Dr. Math

  • (Author)
  • 2005(Publication Date)
  • Jossey-Bass

    (Publisher)

  Yours truly, Quentin Rectangle to Parallel- ogram As you squish a rectangle into a parallelogram, the sides don’t change length, so the perimeter (which is just the sum of the lengths of the sides) must stay the same; but the height decreases, so the area must decrease, too. If you have a set of Parallelograms with the same side lengths, the rectangle will be the one with the largest area. The areas of the oth- ers can get as close to zero as you want by making two of the angles very close to zero and the other two angles very close to 180 degrees. To see why this works, cut off the flaps at the ends of a cereal box. Look through the box to see a rectangle. Now start squishing the box by making it as flat as possible. The perimeter stays the same (since none of the sides are changing length), but the area decreases, like this: These all have the same perimeter but different areas. I’m not entirely sure what your book is talking about—I suspect there are illustrations, which I would need to see—but it sounds to me as though the other possibility is that the base and height stay the same, like this: These all have the same area but different perimeters. From your initial statement, it sounds as though you may be con- fused about the basic ideas of perimeter and area. If that’s the case, you should probably take a look at our first book on geometry, Dr. Math Introduces Geometry. —Dr. Math, The Math Forum Quadrilaterals and Other Polygons 123 Hi, Qian, Let us consider a quadrilateral as you described it: Note that, for instance, KL is parallel to and half the measure of AC, because in triangle ABC, segment KL is a midsegment. The same can be said about MN. So MN and KL are par- allel and congruent. This tells us that KLMN is a parallelogram. Now let X and Y be the inter- sections of AC with KN and LM, respectively. And let E be the intersection of the altitude from B to AC.

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  Prealgebra

  • Alan Tussy, Diane Koenig(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Another type of quadrilateral is the rectangle. Most picture frames and many mirrors are rectangular. Utility knife blades and swimming fins have shapes that are examples of a third type of quadrilateral called a trapezoid. macroworld/iStock/Getty Images Copyright 2019 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. CHAPTER 9 • An Introduction to Geometry 860 We can use the capital letters that label the vertices of a quadrilateral to name it. For example, when referring to the quadrilateral shown on the right, with vertices A, B, C , and D, we can use the notation quadrilateral ABCD. When naming a quadrilateral (or any other polygon), we may begin with any vertex. Then we move around the figure in a clockwise (or counterclockwise) direction as we list the remaining vertices. Some other ways of naming the quadrilateral above are quadrilateral ADCB, quadrilateral CDAB, and quadrilateral DABC . It would be unacceptable to name it as quadrilateral ACDB, because the vertices would not be listed in clockwise (or counterclockwise) order. A segment that joins two nonconsecutive vertices of a polygon is called a diagonal of the polygon. Quadrilateral ABCD shown below has two diagonals, AC and BD . A B D C OBJECTIVE 2 Use properties of rectangles to find unknown angle measures and side lengths. Recall that a rectangle is a quadrilateral with four right angles. The rectangle is probably the most common and recognizable of all geometric figures. For example, most doors and windows are rectangular in shape. The boundaries of soccer fields and basketball courts are rectangles.

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