Rhombuses

9 Key excerpts on "Rhombuses"

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  eBook - PDF

  Introduction to Mathematical Literacy

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

    (Publisher)

  A rhombus is every so often called as a diamond or in the shape of a diamond. The total distance which is traveled end to end to the border of a rhombus is the perimeter of a rhombus. The rhombus can also be called a square because it fulfills all of the requirements or properties the same as compared to a rhombus has. Rhombus has: • All sides of equal length; • Opposite sides have to be parallel; • The altitude is a distance at right angles to two sides; • The diagonals of a rhombus bisect each other at 90 ° . The perimeter of a rhombus is given as: P = 4×a 4.9.2. Area of a Rhombus Formula With the similarity of the diamond, a rhombus is a flat figure which is having the four same straight sides. All of the sides of the rhombus, which is having the same length, opposite sides are parallel to each other, and opposite angles are equal. Another name for rhombus is an equilateral quadrilateral, means that all of its sides are equal in length. A rhombus is in the shape of a slanting square. The area of the rhombus is half the product of the diagonals (Figure 4.20). Perimeter and Area 93 Figure 4.20: Area of the rhombus. The area of a Rhombus formula is: 4 × area of ∆ AOB = 4 × 12 × AO × OB sq. units = 4 × 12 × 12 d 1 × 12 d 2 sq. units = 4 × 18 d 1 × d 2 square units = 12 × d 1 × d 2 In this way, where, Area of a Rhombus = A = 12 × d 1 × d 2 (d 1 and d 2 are the diagonals of the rhombus). 4.10. SQUARE 4.10.1. Area of Square Area of a square is explained as the number of square units which is required to entirely fill a square. Generally, the area is explained as the region which is occupied inside the boundary of a flat object or figure. The evaluation is taking place in the square units with the standard unit being square meters (m 2 ). For the evaluation or determination of the area, there are pre-defined formulas for squares, rectangles, circle, triangles, etc. (Figure 4.21). Introduction to Mathematical Literacy 94 Figure 4.21: Area of the square.

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  Mathematics for Elementary Teachers

  A Contemporary Approach

  • Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  It was established in Example 12.7 that both pairs of opposite sides of a rhombus, rectangle, and a square are parallel. Since the descrip- tion of a parallelogram is a quadrilateral with two pairs of parallel sides, rhombi, rectangles, and squares must also be parallelograms. Because a rhombus, rectangle, and square are all specific types of parallelograms, they have all of the properties of a parallelogram. The listing of the properties of these quadrilaterals shown in Table 12.4 illustrates this point. In the table, the properties of a parallelogram, other than opposite sides being parallel, are that the opposite sides are the same length and the diagonals intersect at their midpoints and therefore bisect each other. A rhombus, rectangle, and square have these same properties since they are parallelograms. We have established how a rhombus, rectangle, and square are all related to paral- lelograms, but we have not discussed how they might be related to each other. The descriptions of these quadrilaterals in Table 12.3 can assist in answering this question. Since a square is a quadrilateral with four sides the same length, we know that it must also be a rhombus. A square is also a quadrilateral with four right angles. Thus, a square must also be a rectangle. A diagram for parallelograms, rhombi, rectangles, and squares, which is similar to Figure 12.39 for triangles, is displayed in Figure 12.40. Parallelograms Squares Rectangles Rhombi Figure 12.40 These relationships in Figure 12.40 can be further seen in Table 12.4 by noticing the common properties. The properties that the rhombus and rectangle have in com- mon are the same properties that each has in common with a parallelogram. The square has all of the properties of a rhombus and all of the properties of a rectangle. It is clear that squares are special types of rectangles and special types of rhombi. Table 12.3 introduced four common quadrilaterals. Table 12.5 introduces three more.

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  eBook - ePub

  Geometry For Dummies

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

    (Publisher)

  Also, note that the rhombus is vertical rather than on its side like parallelograms are usually drawn; this is the easier and better way to draw a rhombus because you can more easily see its symmetry and the fact that its diagonals are perpendicular.) Here are the properties of the rhombus, rectangle, and square. Note that because these three quadrilaterals are all parallelograms, their properties include the parallelogram properties. The rhombus has the following properties: All the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite angles are congruent, and consecutive angles are supplementary). All sides are congruent by definition. The diagonals bisect the angles. The diagonals are perpendicular bisectors of each other. The rectangle has the following properties: All the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite sides are congruent, and diagonals bisect each other). All angles are right angles by definition. The diagonals are congruent. The square has the following properties: All the properties of a rhombus apply (the ones that matter here are parallel sides, diagonals are perpendicular bisectors of each other, and diagonals bisect the angles). All the properties of a rectangle apply (the only one that matters here is diagonals are congruent). All sides are congruent by definition. All angles are right angles by definition. Now try working through a couple of problems: Given the rectangle as shown, find the measures of and : © John Wiley & Sons, Inc. Here’s the solution: MNPQ is a rectangle, so. Thus, because there are in a triangle, you can say the following: Now plug in 14 for all the x ’s. Angle QMP,, is, or, and because you have a rectangle, is the complement of and is therefore, or

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  eBook - ePub

  GED® Math Test Tutor, 2nd Edition

  • Sandra Rush(Author)
  • 2017(Publication Date)
  • Research & Education Association

    (Publisher)

  supplementary angles.

  Rhombus

  If we add the condition that all four sides of a parallelogram are equal, then we have a rhombus. So a rhombus has all of the properties of a parallelogram plus the sides are equal.

  Thus, in the above figure of the rhombus, PQ = QR = RS = SP = s, and the perimeter can be written as

  Likewise, the area of the rhombus is

  where any of the sides can be used as the base, and the height drawn to each side is the same. The diagonals of a rhombus bisect each other (as they did for the parallelogram), but now they also are perpendicular to each other.

  Rectangle

  If, instead of saying the four sides of the parallelogram are equal, we say that the four angles are equal, we have a rectangle, which is a parallelogram with four equal angles.

  Thus, in the figure of the rectangle above, ∠T= ∠U = ∠V= ∠W, and since the angles of a quadrilateral add up to 360°, each of the four angles is 90°, or a right angle.

  The opposite sides are equal, as in a parallelogram, but not all sides are equal (as they were in the rhombus). The perimeter is written as usual as

  Because all the angles are right angles, all sides l (length) are perpendicular to sides w (width), so they take the place of the base and height, and the area of the rectangle is

  HINT

  If we think of tiling a floor in a straight line, we get an idea of why area is length times width. Suppose we wanted to tile a room that is 12 feet by 10 feet in 1 -foot tiles. If you count the number of tiles needed, it would be 12 rows of 10 tiles (or 10 rows of 12 tiles), and that total is 120 tiles, each 1 foot square, which is the area of the floor: 120 square feet.

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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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  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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  Prealgebra

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

    (Publisher)

  Properties of rectangles: 1. All four angles are right angles. 2. Opposite sides are parallel. 3. Opposite sides have equal length. 4. Diagonals have equal length. 5. The diagonals intersect at their midpoints. Rectangle ABCD 16 in. 17 in. 30 in. B E A D C 1. m(DAB ) 5 m(ABC ) 5 m(BCD ) 5 m(CDA ) 5 90° 2. AD  BC and AB  DC 3. m(AD ) 5 16 in. and m(DC ) 5 30 in. 4. m(DB ) 5 m(AC ) 5 34 in. 5. m(DE ) 5 m(AE ) 5 m(EC ) 5 17 in. Conditions that a parallelogram must meet to ensure that it is a rectangle: 1. If a parallelogram has one right angle, then the parallelogram is a rectangle. 2. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Read Example 2 on page 861 to see how these two conditions are used in construction to “square a foundation.” 12 ft 12 ft 9 ft 9 ft A B D C A trapezoid is a quadrilateral with exactly two sides parallel. The parallel sides of a trapezoid are called bases. The nonparallel sides are called legs. If the legs (the nonparallel sides) of a trapezoid are of equal length, it is called an isosceles trapezoid. In an isosceles trapezoid, both pairs of base angles are congruent. Trapezoid ABCD A B D C Upper base Lower base Lower base angles Upper base angles Leg Leg AB DC || The sum S, in degrees, of the measures of the angles of a polygon with n sides is given by the formula S 5 (n 2 2)180° Find the sum of the angle measures of a hexagon. Since a hexagon has six sides, we will substitute 6 for n in the formula. S 5 (n 2 2)180° S 5 (6 2 2)180° Substitute 6 for n, the number of sides. S 5 (4)180° Do the subtraction within the parentheses. S 5 720° Do the multiplication. The sum of the measures of the angles of a hexagon is 720°. 9 • Summary and Review 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).

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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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  The Essence of Mathematics Through Elementary Problems

  • Tony Gardiner, Alexandre Borovik(Authors)
  • 2019(Publication Date)
  • Open Book Publishers

    (Publisher)

  A quadrilateral ABCD in which AB bardbl DC and BC bardbl AD is called a parallelogram . A parallelogram ABCD with a right angle is a rectangle . A parallelogram ABCD with AB “ AD is called a rhombus . A rectangle which is also a rhombus is called a square . Problem 157 Let ABCD be a parallelogram. (i) Prove that △ ABC ” △ CDA , so that each triangle has area exactly half of area( ABCD ). (ii) Conclude that opposite sides of ABCD are equal in pairs and that opposite angles are equal in pairs. (iii) Let AC and BD meet at X . Prove that X is the midpoint of both AC and BD . △ Problem 158 Let ABCD be a parallelogram with centre X (where the two main diagonals AC and BD meet), and let m be any straight line passing through the centre. Prove that m divides the parallelogram into two parts of equal area. △ We defined a parallelogram to be “a quadrilateral ABCD in which AB bardbl DC and BC bardbl AD ”; however, in practice, we need to be able to recognise a parallelogram even if it is not presented in this form. The next result hints at the variety of other conditions which allow us to recognise a given quadrilateral as being a parallelogram “in mild disguise”. Problem 159 (a) Let ABCD be a quadrilateral in which AB bardbl DC , and AB “ DC . Prove that BC bardbl AD , and hence that ABCD is a parallelogram. (b) Let ABCD be a quadrilateral in which AB “ DC and BC “ AD . Prove that AB bardbl DC , and hence that ABCD is a parallelogram. (c) Let ABCD be a quadrilateral in which = A “ = C and = B “ = D . Prove that AB bardbl DC and that BC bardbl AD , and hence that ABCD is a parallelogram. △ The next problem presents a single illustrative example of the kinds of things which we know in our bones must be true, but where the reason, or proof, may need a little thought. 188 Geometry Problem 160 Let ABCD be a parallelogram. Let M be the midpoint of AD and N be the midpoint of BC .

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