Mathematics
Trapezoids
A trapezoid is a quadrilateral with at least one pair of parallel sides. It is also known as a trapezium in some countries. The parallel sides are called the bases, and the non-parallel sides are the legs. The area of a trapezoid can be calculated using the formula A = (1/2)h(b1 + b2), where h is the height and b1 and b2 are the lengths of the bases.
Written by Perlego with AI-assistance
Related key terms
1 of 5
6 Key excerpts on "Trapezoids"
eBook - PDF- Alberto D. Yazon(Author)
- 2019(Publication Date)
- Society Publishing(Publisher)
4.11.1.1. Perimeter The type of polygon that is nothing but a quadrilateral which is having at least a pair of parallel sides is known as the Trapezoid. Two sides of a trapezium are parallel. The perpendicular distance between the parallel sides is known as the altitude (Figure 4.23). Figure 4.23: Perimeter of a trapezoid. Introduction to Mathematical Literacy 96 The perimeter of a trapezoid formula can be written in the form of: P = a + b + c + d where, a, b, c, d are the lengths of each side. 4.11.2. Area of Trapezoid Trapezium area can be evaluated with the help of the application of the formula which is given below: Area = (1/2) h (AB + CD) Form the above figure, the Trapezium perimeter formula is as follows: Perimeter = Sum of all the sides = AB + BC + CD + DA 4.12. POLYGON A polygon which is having equal sides’ that is equilateral and equal angles that is equiangular is known as a regular polygon. An apothem is applied to determine or evaluate the area of a regular polygon. Apothem is a segment that links or connects the center of the polygon to the midpoint of any of the side, and it is perpendicular with respect to that side (Figure 4.24). Figure 4.24: A block diagram of the polygon. Source: https://live.staticflickr.com/5589/14779107992_ab3475d26b_b.jpg. All of the vertices of a regular polygon lie on a common circle (which is the circumscribed circle), that is they are concyclic points. That is, a regular polygon is a cyclic polygon. Along with it, the property of sides which are having the equal sides, this shows that every of the regular polygon also has an inscribed circle or in circle that is tangent to every side at the midpoint. Perimeter and Area 97 In this way, a regular polygon is a tangential polygon. A regular n-sided polygon can be structured with the help of the compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes.
eBook - PDFDr. Math Introduces Geometry
Learning Geometry is Easy! Just ask Dr. Math!
- (Author)
- 2004(Publication Date)
- Jossey-Bass(Publisher)
4. A parallelogram is a quadrilateral with exactly two pairs of parallel sides. Every rhombus is a parallelogram and so is every rectangle. And if every rectangle is a parallelogram, then so is every square. 5. There are two definitions commonly used for trapezoid. The traditional American definition is a quadrilateral with exactly one pair of parallel sides. The British and “new” American definition is a quadrilateral with at least one pair of parallel sides. In this book we will use the second definition, which means that all parallelograms (including rhombuses, rectan- gles, and squares) may be considered Trapezoids, because they all have at least one pair of parallel sides. (If the trapezoid is isosceles, then the nonparallel sides have the same length and the base angles are equal.) 6. A kite may or may not have parallel sides; it does have two pairs of adjacent sides with equal lengths—that is, instead of being across from each other, the sides with equal lengths are next to each other. So a kite can look like the kind of toy you’d fly in a field on a windy day. But a rhombus and a square are also special cases of a kite: while they do have two pairs of adjacent sides that have equal lengths, those lengths are also equal to each other. Just as there are two definitions for the trapezoid, there are two definitions for the kite. We use the one given above; some people use one that says the two pairs of congruent sides must have different lengths, so for them, a rhombus (and therefore a square) is not a kite. 7. A scalene quadrilateral has four unequal sides that are not parallel. —Dr. Math, The Math Forum Dear Leon, Your definition of a kite seems awkward. The people who wrote the definition want to make sure you don’t count three consecutive con- gruent sides as two pairs, so they say you can’t use the same side twice. I can’t imagine why they bother saying “at least two pairs,” since once you’ve chosen two separate pairs, you’ve used up all the sides.
eBook - PDF- Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
43. Two sets of rails (railroad tracks are equally spaced) intersect but not at right angles. Being as specific as possible, indicate what type of quadrilateral WXYZ is formed. 44. In square ABCD (not shown), point E lies on side DC . If AB 5 8 and AE 5 10, find BE. 45. In square ABCD (not shown), point E lies in the interior of ABCD in such a way that uni25B3ABE is an equilateral triangle. Find muni2220DEC. 46. The sides of square ABCD are trisected at the indicated points. If AB 5 3, find the perimeter of (a) quadrilateral EGIK. (b) quadrilateral EHIL. D C A B J K L H G I E F W X Y Z KEY CONCEPTS Trapezoid Bases Legs Base Angles Median Isosceles Trapezoid Right Trapezoid Altitude of a Trapezoid 4.4 The Trapezoid A trapezoid is a quadrilateral with exactly two parallel sides. DEFINITION Consider Figure 4.32. If HL y JK , then HJKL is a trapezoid. The parallel sides HL and JK of trapezoid HJKL are its bases, and the nonparallel sides HJ and LK are its legs. Because uni2220J and uni2220K both have base JK for a side, they are a pair of base angles of the trapezoid; uni2220H and uni2220L are also a pair of base angles because HL is a base. When the midpoints of the two legs of a trapezoid are joined, the resulting line segment is known as the median of the trapezoid. [See Figure 4.33(a)]. Given that M and N are the midpoints of the legs HJ and LK in trapezoid HJKL, MN is the median of the trapezoid. If the two legs of a trapezoid are congruent, the trapezoid is known as an isosceles trapezoid. In Figure 4.33(b), RSTV is an isosceles trapezoid because RS y VT and RV _ ST . J K L H base base leg leg Figure 4.32 K J H L M N (a) median T V R S (b) K J H L (c) Y Z W X (d) Figure 4.33 4.4 ■ The Trapezoid 207 Every trapezoid contains two pairs of consecutive interior angles that are supplemen- tary. Each of these pairs of angles is formed when parallel lines are cut by a transversal.
eBook - ePubMaking Sense of Mathematics for Teaching Grades 6-8
(Unifying Topics for an Understanding of Functions, Statistics, and Probability)
- Edward C. Nolan, Juli K. Dixon(Authors)
- 2016(Publication Date)
- Solution Tree Press(Publisher)
You also engaged in Mathematical Practice 7, “Look for and make use of structure,” in that you used the structure and properties of shapes to make sense of finding the area of an unknown shape, in this case, a trapezoid. By exploring the geometric structure of the trapezoid, the areas of rectangles, triangles, and parallelograms were used to make sense of the area of a trapezoid. The structure of equations and expressions also supported your viable arguments when you found equivalent expressions. This use of other topics in mathematics helps reinforce the connections between ideas in mathematics and the progression of those ideas throughout the grades.An Alternate Method
Consider one more method for determining the area of this trapezoid (see figure 5.8 ). In this case, the trapezoid is transformed into a rectangle. How is this rectangle created? How can you determine the base and height of this rectangle?Figure 5.8: Transforming a trapezoid into a rectangle.The rectangle is created by first decomposing the trapezoid into a rectangle and two triangles similar to the decomposition in figure 5.3 (page 106 ). Next, the base of each decomposed triangle is bisected and two new smaller triangles are formed. These triangles are rotated 180° about the midpoint of the hypotenuse for each original triangle. The rotated smaller triangles form rectangles with the part of the triangles that were not rotated. These rectangles are composed with the rectangle from the trapezoid to make a larger rectangle. How, then, is the length of the newly formed base of the rectangle determined? Since you have rotated part of the base of each triangle to the opposite base of the trapezoid, the length of the opposite sides of the newly formed rectangle is the mean of the two bases of the original trapezoid, or ½(b 1 + b 2 ). Using the formula for the area of a rectangle leads to the formulaAtrap= ½(b 1 + b 2 )h . This strategy provides another way that you can make sense of the formula for the area of a trapezoid.The Progression
The formal development of understanding of measurement and geometry begins in kindergarten with students making sense of shapes and spatial relationships. As students progress throughout the elementary grades, the complexity and type of shapes change. In the middle grades, students apply the properties learned in the elementary grades to a variety of more complex shapes and in new ways. Following is a progression within measurement and geometry in grades 6–8.
- Allen Ma, Amber Kuang(Authors)
- 2022(Publication Date)
- For Dummies(Publisher)
A parallelogram is a quadrilateral with two pairs of congruent sides and two pairs of congruent angles. A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram in which all sides are congruent. A square has the properties of both a rhombus and a rectangle, which means it has four congruent sides and four right angles. A trapezoid is a quadrilateral with only one pair of parallel sides. An isosceles trapezoid is a trapezoid in which the nonparallel sides are congruent. This chapter focuses on the similarities and differences between these quadrilaterals and applies them toward algebraic problems. The Problems You’ll Work On In this chapter, you see a variety of geometry problems: » Understanding the properties of a parallelogram » Understanding the properties of a rectangle » Understanding the properties of a rhombus » Understanding the properties of a square » Understanding the properties of a trapezoid What to Watch Out For To avoid common mistakes, try the following tips: » Be careful when naming a quadrilateral by its vertices. The letters must go in consecutive order. » Because these quadrilaterals have parallel sides, keep an eye out for alternate interior angles formed when the diagonal acts as the transversal. » Remember that the diagonals of a parallelogram do not bisect the angles of the parallelogram. » Make sure you understand what the question is asking you to solve for. Chapter 11 90 PART 1 The Questions Properties of Parallelograms 512–525 In parallelogram MATH, diagonals MT and AH intersect at E. Use the figure and the given information to solve each problem. 512. If AT x 8 2 and MH x 5 8 , find the length of MH . 513. If AM 40 and TH x 2 10 , find the value of x . 514. If m AMH 80 and m HTA is represented by x 50 , find the value of x . 515. If m AMH is represented by x 40 and m MAT 110 , find the measure of AMH . 516. If m ATH and m MHT are represented by 2 25 x and 3 5 x , respectively, find the measure of MHT .
eBook - ePub- Mark Ryan(Author)
- 2016(Publication Date)
- For Dummies(Publisher)
Due to space considerations, I’m going to skip the game plan this time. You’re on your own. © John Wiley & Sons, Inc. © John Wiley & Sons, Inc. Properties of the trapezoid and the isosceles trapezoid Practice your picking-out-properties proficiency one more time with the trapezoid and isosceles trapezoid in Figure 10-10. Remember: What looks true is likely true, and what doesn’t, isn’t. The properties of the trapezoid are as follows: The bases are parallel by definition. Each lower base angle is supplementary to the upper base angle on the same side. The properties of the isosceles Trapezoids are as follows: The properties of Trapezoids apply by definition (parallel bases). The legs are congruent by definition. The lower base angles are congruent. The upper base angles are congruent. Any lower base angle is supplementary to any upper base angle. The diagonals are congruent. © John Wiley & Sons, Inc. FIGURE 10-10: A trapezoid (on the left) and an isosceles trapezoid (on the right). Perhaps the hardest property to spot in both diagrams is the one about supplementary angles. Because of the parallel sides, consecutive angles are same-side interior angles and are thus supplementary. (All the quadrilaterals except the kite, by the way, contain consecutive supplementary angles.) Here’s an isosceles trapezoid proof for you. I trust you to handle the game plan on your own again. © John Wiley & Sons, Inc. © John Wiley & Sons, Inc. Chapter 11 Proving That You Have a Particular Quadrilateral IN THIS CHAPTER Noting the property-proof connection Proving a figure is a parallelogram or other quadrilateral Chapter 10 tells you all about seven different quadrilaterals — their definitions, their properties, what they look like, and where they fit on the quadrilateral family tree
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
