DOCSLIB.ORG

Areas of Triangles and Trapezoids

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Areas of Triangles and Trapezoids N S O S E L Areas of Triangles and Trapezoids Goal: Find the areas of triangles and trapezoids. Vocabulary Base of The base of a triangle is the length of any one of a triangle: its sides. The height of a triangle is the perpendicular distance Height of between the side whose length is the base and the a triangle: vertex opposite that side. Bases of a The lengths of the parallel sides of a trapezoid are trapezoid: the bases of a trapezoid. Height of The height of a trapezoid is the perpendicular a trapezoid: distance between the bases. Area of a Triangle Words The area A of a triangle is half the product of a base and the corresponding height . height h Algebra ϭ ᎏ1ᎏ base b A 2bh EXAMPLE 1 Finding the Area of a Triangle Sculpture An artist is creating a sculpture that includes a triangular face that has a base 125 feet long and a height of 84 feet. Find the area of the triangular face. 84 ft Solution Write formula for area A ϭ ᎏ1ᎏbh 2 of a triangle. 125 ft ϭ ᎏ1ᎏ 125 84 2 ΂΃΂΃Substitute 125 for b and 84 for h. ϭ 5250 Multiply. Answer: The area of the face is 5250 square feet . Lesson 11.5 Areas of Triangles and Trapezoids | 251 EXAMPLE 2 Finding the Base of a Triangle A triangle has a height of 15 centimeters and an area of 202.5 square centimeters. Find the base of the triangle. 1 A ϭ ᎏᎏbh Write formula for area of a triangle. 2 1 202.5 ϭ ᎏᎏb(15) Substitute 202.5 for A and 15 for h. 2 202.5 ϭ 7.5b Simplify. 27 ϭ b Divide each side by 7.5 . Answer: The base of the triangle is 27 centimeters . Guided Practice Find the unknown area or height of the triangle. 1. A ___,? b ϭ 7 ft, h ϭ 12 ft 2. A ϭ 52 m2, b ϭ 8 m, h ___? 42 ft2 13 m Because a trapezoid has more Area of a Trapezoid than one base, the Words The area A of a trapezoid is base b bases of a trapezoid are half 1 b usually labeled 1 the product of the sum of the bases and b b and 2. 1 is read height h "b sub one." theheight . base b2 1 Algebra A ϭ ᎏᎏ(b ϩ b )h 2 1 2 252 | Chapter 11 Notetaking Guide EXAMPLE 3 Finding the Area of a Trapezoid Find the area of the trapezoid shown. 8 in. ϭ ᎏ1ᎏ ϩ A 2(b1 b2)h Write formula for area of a trapezoid. 7 in. ϭ ᎏ1ᎏ ϩ 2 ΂ 8 3 ΃΂7 ΃ Substitute 8 for b1, 3 3 in. for b2, and 7 for h. ϭ 38.5 Simplify. Answer: The area of the trapezoid is 38.5 square inches . EXAMPLE 4 Finding the Height of a Trapezoid A trapezoid has an area of 82 square meters. The bases are 11 meters and 9 meters. Find the height. ϭ ᎏ1ᎏ ϩ A 2(b1 b2)h Write formula for area of a trapezoid. Substitute 82 for A,11 for b1, ϭ ᎏ1ᎏ ϩ 82 2 ΂ 11 9 ΃h and9 for b2. ϭ ᎏ1ᎏ 82 2 ΂΃20 h Add. 82 ϭ 10 h Multiply. 8.2 ϭ h Divide each side by 10 . Answer: The height of the trapezoid is 8.2 meters . Guided Practice Find the unknown area, base, or height of the trapezoid. ϭ ϭ ϭ 3. A ___,? b1 12 ft, b2 8 ft, h 4 ft 40 ft2 2 ϭ ϭ ϭ 4. A 22 m , b1 7 m, b2 ___,? h 4 m 4 m 2 ϭ ϭ ϭ 5. A 15 cm , b1 3 cm, b2 7 cm, h ___? 3 cm Lesson 11.5 Areas of Triangles and Trapezoids | 253.
Load more

Recommended publications
  • Square Rectangle Triangle Diamond (Rhombus) Oval Cylinder Octagon Pentagon Cone Cube Hexagon Pyramid Sphere Star Circle
    SQUARE RECTANGLE TRIANGLE DIAMOND (RHOMBUS) OVAL CYLINDER OCTAGON PENTAGON CONE CUBE HEXAGON PYRAMID SPHERE STAR CIRCLE Powered by: www.mymathtables.com Page 1 what is Rectangle? • A rectangle is a four-sided flat shape where every angle is a right angle (90°). means "right angle" and show equal sides. what is Triangle? • A triangle is a polygon with three edges and three vertices. what is Octagon? • An octagon (eight angles) is an eight-sided polygon or eight-gon. what is Hexagon? • a hexagon is a six-sided polygon or six-gon. The total of the internal angles of any hexagon is 720°. what is Pentagon? • a plane figure with five straight sides and five angles. what is Square? • a plane figure with four equal straight sides and four right angles. • every angle is a right angle (90°) means "right ang le" show equal sides. what is Rhombus? • is a flat shape with four equal straight sides. A rhombus looks like a diamond. All sides have equal length. Opposite sides are parallel, and opposite angles are equal what is Oval? • Many distinct curves are commonly called ovals or are said to have an "oval shape". • Generally, to be called an oval, a plane curve should resemble the outline of an egg or an ellipse. Powered by: www.mymathtables.com Page 2 What is Cube? • Six equal square faces.tweleve edges and eight vertices • the angle between two adjacent faces is ninety. what is Sphere? • no faces,sides,vertices • All points are located at the same distance from the center. what is Cylinder? • two circular faces that are congruent and parallel • faces connected by a curved surface.
    [Show full text]
  • Applying the Polygon Angle
    POLYGONS 8.1.1 – 8.1.5 After studying triangles and quadrilaterals, students now extend their study to all polygons. A polygon is a closed, two-dimensional figure made of three or more non- intersecting straight line segments connected end-to-end. Using the fact that the sum of the measures of the angles in a triangle is 180°, students learn a method to determine the sum of the measures of the interior angles of any polygon. Next they explore the sum of the measures of the exterior angles of a polygon. Finally they use the information about the angles of polygons along with their Triangle Toolkits to find the areas of regular polygons. See the Math Notes boxes in Lessons 8.1.1, 8.1.5, and 8.3.1. Example 1 4x + 7 3x + 1 x + 1 The figure at right is a hexagon. What is the sum of the measures of the interior angles of a hexagon? Explain how you know. Then write an equation and solve for x. 2x 3x – 5 5x – 4 One way to find the sum of the interior angles of the 9 hexagon is to divide the figure into triangles. There are 11 several different ways to do this, but keep in mind that we 8 are trying to add the interior angles at the vertices. One 6 12 way to divide the hexagon into triangles is to draw in all of 10 the diagonals from a single vertex, as shown at right. 7 Doing this forms four triangles, each with angle measures 5 4 3 1 summing to 180°.
    [Show full text]
  • Petrie Schemes
    Canad. J. Math. Vol. 57 (4), 2005 pp. 844–870 Petrie Schemes Gordon Williams Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grunbaum–Dress¨ polyhedra, pos- sess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. 1 Introduction Historically, polyhedra have been conceived of either as closed surfaces (usually topo- logical spheres) made up of planar polygons joined edge to edge or as solids enclosed by such a surface. In recent times, mathematicians have considered polyhedra to be convex polytopes, simplicial spheres, or combinatorial structures such as abstract polytopes or incidence complexes. A Petrie polygon of a polyhedron is a sequence of edges of the polyhedron where any two consecutive elements of the sequence have a vertex and face in common, but no three consecutive edges share a commonface. For the regular polyhedra, the Petrie polygons form the equatorial skew polygons. Petrie polygons may be defined analogously for polytopes as well. Petrie polygons have been very useful in the study of polyhedra and polytopes, especially regular polytopes.
    [Show full text]
  • Right Triangles and the Pythagorean Theorem Related?
    Activity Assess 9-6 EXPLORE & REASON Right Triangles and Consider △​ ABC​ with altitude ​​CD‾ ​​ as shown. the Pythagorean B Theorem D PearsonRealize.com A 45 C 5√2 I CAN… prove the Pythagorean Theorem using A. What is the area of △​ ​ABC? ​Of △​ACD? Explain your answers. similarity and establish the relationships in special right B. Find the lengths of ​​AD‾ ​​ and ​​AB‾ ​​. triangles. C. Look for Relationships Divide the length of the hypotenuse of △​ ABC​ VOCABULARY by the length of one of its sides. Divide the length of the hypotenuse of ​ △ACD​ by the length of one of its sides. Make a conjecture that explains • Pythagorean triple the results. ESSENTIAL QUESTION How are similarity in right triangles and the Pythagorean Theorem related? Remember that the Pythagorean Theorem and its converse describe how the side lengths of right triangles are related. THEOREM 9-8 Pythagorean Theorem If a triangle is a right triangle, If... ​△ABC​ is a right triangle. then the sum of the squares of the B lengths of the legs is equal to the square of the length of the hypotenuse. c a A C b 2 2 2 PROOF: SEE EXAMPLE 1. Then... ​​a​​ ​ + ​b​​ ​ = ​c​​ ​ THEOREM 9-9 Converse of the Pythagorean Theorem 2 2 2 If the sum of the squares of the If... ​​a​​ ​ + ​b​​ ​ = ​c​​ ​ lengths of two sides of a triangle is B equal to the square of the length of the third side, then the triangle is a right triangle. c a A C b PROOF: SEE EXERCISE 17. Then... ​△ABC​ is a right triangle.
    [Show full text]
  • Cyclic Quadrilaterals — the Big Picture Yufei Zhao [email protected]
    Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao Cyclic Quadrilaterals | The Big Picture Yufei Zhao [email protected] An important skill of an olympiad geometer is being able to recognize known configurations. Indeed, many geometry problems are built on a few common themes. In this lecture, we will explore one such configuration. 1 What Do These Problems Have in Common? 1. (IMO 1985) A circle with center O passes through the vertices A and C of triangle ABC and intersects segments AB and BC again at distinct points K and N, respectively. The circumcircles of triangles ABC and KBN intersects at exactly two distinct points B and M. ◦ Prove that \OMB = 90 . B M N K O A C 2. (Russia 1995; Romanian TST 1996; Iran 1997) Consider a circle with diameter AB and center O, and let C and D be two points on this circle. The line CD meets the line AB at a point M satisfying MB < MA and MD < MC. Let K be the point of intersection (different from ◦ O) of the circumcircles of triangles AOC and DOB. Show that \MKO = 90 . C D K M A O B 3. (USA TST 2007) Triangle ABC is inscribed in circle !. The tangent lines to ! at B and C meet at T . Point S lies on ray BC such that AS ? AT . Points B1 and C1 lies on ray ST (with C1 in between B1 and S) such that B1T = BT = C1T . Prove that triangles ABC and AB1C1 are similar to each other. 1 Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao A B S C C1 B1 T Although these geometric configurations may seem very different at first sight, they are actually very related.
    [Show full text]
  • Cyclic Quadrilaterals
    GI_PAGES19-42 3/13/03 7:02 PM Page 1 Cyclic Quadrilaterals Definition: Cyclic quadrilateral—a quadrilateral inscribed in a circle (Figure 1). Construct and Investigate: 1. Construct a circle on the Voyage™ 200 with Cabri screen, and label its center O. Using the Polygon tool, construct quadrilateral ABCD where A, B, C, and D are on circle O. By the definition given Figure 1 above, ABCD is a cyclic quadrilateral (Figure 1). Cyclic quadrilaterals have many interesting and surprising properties. Use the Voyage 200 with Cabri tools to investigate the properties of cyclic quadrilateral ABCD. See whether you can discover several relationships that appear to be true regardless of the size of the circle or the location of A, B, C, and D on the circle. 2. Measure the lengths of the sides and diagonals of quadrilateral ABCD. See whether you can discover a relationship that is always true of these six measurements for all cyclic quadrilaterals. This relationship has been known for 1800 years and is called Ptolemy’s Theorem after Alexandrian mathematician Claudius Ptolemaeus (A.D. 85 to 165). 3. Determine which quadrilaterals from the quadrilateral hierarchy can be cyclic quadrilaterals (Figure 2). 4. Over 1300 years ago, the Hindu mathematician Brahmagupta discovered that the area of a cyclic Figure 2 quadrilateral can be determined by the formula: A = (s – a)(s – b)(s – c)(s – d) where a, b, c, and d are the lengths of the sides of the a + b + c + d quadrilateral and s is the semiperimeter given by s = 2 . Using cyclic quadrilaterals, verify these relationships.
    [Show full text]
  • Lessons Designed to Teach Fourth Grade Students the Concept Equilateral Triangle at the Formal Level of Attainment* Practical Paper No
    DOCUMENT RESUME ED 100 720 SE 018 753 AUTHOR McMurray, Nancy E.; And Others TITLE Lessons Designed to Teach Fourth Grade Students the Concept Equilateral Triangle at the Formal Level of Attainment* Practical Paper No. 14. INSTITUTION Wisconsin Univ., Madison. Research and Development Center for Cognitive Learning. SPONS AGENCY National Inst. of Education (DREW), Washington, D.C. REPORT NO WRDCCL-PP-14 PUB DATE Sep 74 CONTRACT NE-C-00-3-0065 NOTE 56p.; Report from the Project on Conditions of School Learning and Instructional Strategies EARS PRICE MF-$0.75 HC-$3.15 PLUS POSTAGE DESCRIPTORS *Concept Formation; *Elementary School Mathematics; *Geometric Concepts; Geometry; Individual Instruction; *instructional Materials; Learning Activities; Learning Theories; *Mathematical Vocabulary; Research; Worksheets IDENTIFIERS Equilateral Triangle ABSTRACT In the first of these two lessons, student study the concepts of three sides of equal length, three equal angles, plane figure, closed figure, and simple figure by reading descriptions, considering examples and nonexamples, and completing exercises. On the second day they combine these concepts in a definition of equilateral triangle. The materials are self-contained but should be preceded by introduction or review of a vocabulary list provided; answers and explanations immediately follow each set of exercises. A brief introductory review of research related to concept learning describes the rationale for the techniques used within the lessons which were developed as part of the Project on Conditions of School Learning and Instructional Strategies at the University of Wisconsin. (SD) i U S DEPARTMENT OF HEALTH, EDUCATION & WELFARE NATIONAL. INSTITUTE OF EDUCATION THIS DOCUMENT HAS BEEN REPRO DUCE.D EXACT,v AS RECEIVED FROM THE PERSON OR uRC,..NiZATION ORIGIN A TIN.; IT POINTS OF viEW OR OPINIONS STA t CO DO NOT NECESSARILY REPRE SENT OFFICIAL NATIONAL INSTITUTE OF EDUCATION POSITION OR POLICY k PE rtmissiori TOlit pr+outla.THIS COPY.
    [Show full text]
  • Two-Dimensional Figures a Plane Is a Flat Surface That Extends Infinitely in All Directions
    NAME CLASS DATE Two-Dimensional Figures A plane is a flat surface that extends infinitely in all directions. A parallelogram like the one below is often used to model a plane, but remember that a plane—unlike a parallelogram—has no boundaries or sides. A plane figure or two-dimensional figure is a figure that lies completely in one plane. When you draw, either by hand or with a computer program, you draw two-dimensional figures. Blueprints are two-dimensional models of real-life objects. Polygons are closed, two-dimensional figures formed by three or more line segments that intersect only at their endpoints. These figures are polygons. These figures are not polygons. This is not a polygon A heart is not a polygon A circle is not a polygon because it is an open because it is has curves. because it is made of figure. a curve. Polygons are named by the number of sides and angles they have. A polygon always has the same number of sides as angles. Listed on the next page are the most common polygons. Each of the polygons shown is a regular polygon. All the angles of a regular polygon have the same measure and all the sides are the same length. SpringBoard® Course 1 Math Skills Workshop 89 Unit 5 • Getting Ready Practice MSW_C1_SE.indb 89 20/07/19 1:05 PM Two-Dimensional Figures (continued) Triangle Quadrilateral Pentagon Hexagon 3 sides; 3 angles 4 sides; 4 angles 5 sides; 5 angles 6 sides; 6 angles Heptagon Octagon Nonagon Decagon 7 sides; 7 angles 8 sides; 8 angles 9 sides; 9 angles 10 sides; 10 angles EXAMPLE A Classify the polygon.
    [Show full text]
  • Architectural Elements Building Strong Shapes
    ARCHITECTURAL ELEMENTS BUILDING STRONG SHAPES What is the strongest geometric shape? There are several shapes that are used when strength is important. THE CIRCLE The arc (think: circle) is the strongest structural shape, and in nature, the sphere is the strongest 3-d shape. The reason being is that stress is distributed equally along the arc instead of concentrating at any one point. Storage silos, storage tanks, diving helmets, space helmets, gas tanks, bubbles, planets, etc. use cylinder or sphere shapes -- or both. THE TRIANGLE The triangle is the strongest to as it holds it shape and has a base which is very strong a also has a strong support. The triangle is common in all sorts of building supports and trusses. It is strong because the three legs of a triangle define one and only one triangle. If all three sides are made of rigid material, the angles are fixed and cannot get larger or smaller without breaking at the joints, unlike a rectangle, for example, which can turn into a parallelogram and even collapse totally. If you take a rectangle and place one diagonal piece from corner to corner, you can make that strong and stable, too, but doing that makes two triangles!! Think about it! so yes, it is the strongest shape THE CATENARY CURVE The overall shape of many bridges is in the shape of a catenary curve. The catenary curve is the strongest shape for an arch which supports only its own shape. Freely hanging cables naturally form a catenary curve. THE HEXAGON The hexagon is the strongest shape known.
    [Show full text]
  • Hexagon Fractions
    Hexagon Fractions A hexagon can be divided up into several equal, different-sized parts. For example, if we draw a line across the center, like this: We have separated the hexagon into two equal-sized parts. Each one is one-half of the hexagon. We could draw the lines differently: Here we have separated the hexagon into three equal-sized parts. Each one of these is one-third of the hexagon. 1 Another way of separating the hexagon into equal-sized parts looks like this: There are six equal-sized parts here, so each one is one-sixth of the hexagon. Each of these shapes has a special name. The shape that is one- half of the hexagon is called a trapezoid: The one that is one-third of the hexagon is a rhombus: 2 and the one that is one-sixth of the hexagon is a … Yes, a triangle! You knew that one, for sure. We can use these shapes to get an understanding of how fractions are added and subtracted. Remember that the numerator of a fraction is just a regular number. The denominator of a fraction is more than just a number, it is sort of a “thing,” like apple, dog, or flower. But it is a “numerical thing” and has a name. We should think of “half” or “third” or “tenth” or whatever as a name, just like “apple” or “dog” or “flower” is the name of something. The numerator in a fraction tells us how many of the denominators we have. For example, because the rhombus is one-third of a hexagon, we can think of the fraction “one-third” as “one rhombus.” For a hexagon, “third” and “rhombus” are two names for the same thing.
    [Show full text]
  • MYSTERIES of the EQUILATERAL TRIANGLE, First Published 2010
    MYSTERIES OF THE EQUILATERAL TRIANGLE Brian J. McCartin Applied Mathematics Kettering University HIKARI LT D HIKARI LTD Hikari Ltd is a publisher of international scientific journals and books. www.m-hikari.com Brian J. McCartin, MYSTERIES OF THE EQUILATERAL TRIANGLE, First published 2010. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission of the publisher Hikari Ltd. ISBN 978-954-91999-5-6 Copyright c 2010 by Brian J. McCartin Typeset using LATEX. Mathematics Subject Classification: 00A08, 00A09, 00A69, 01A05, 01A70, 51M04, 97U40 Keywords: equilateral triangle, history of mathematics, mathematical bi- ography, recreational mathematics, mathematics competitions, applied math- ematics Published by Hikari Ltd Dedicated to our beloved Beta Katzenteufel for completing our equilateral triangle. Euclid and the Equilateral Triangle (Elements: Book I, Proposition 1) Preface v PREFACE Welcome to Mysteries of the Equilateral Triangle (MOTET), my collection of equilateral triangular arcana. While at first sight this might seem an id- iosyncratic choice of subject matter for such a detailed and elaborate study, a moment’s reflection reveals the worthiness of its selection. Human beings, “being as they be”, tend to take for granted some of their greatest discoveries (witness the wheel, fire, language, music,...). In Mathe- matics, the once flourishing topic of Triangle Geometry has turned fallow and fallen out of vogue (although Phil Davis offers us hope that it may be resusci- tated by The Computer [70]). A regrettable casualty of this general decline in prominence has been the Equilateral Triangle. Yet, the facts remain that Mathematics resides at the very core of human civilization, Geometry lies at the structural heart of Mathematics and the Equilateral Triangle provides one of the marble pillars of Geometry.
    [Show full text]
  • The Triangle Square up Ruler Use This Ruler to Square Pieces up to 9H"
    The Triangle Square Up Ruler Use this ruler to square pieces up to 9H". Finished Cut Square Finished Cut Square Size Squares Up To Size Squares Up To H" 3" 1" 1" 4" 1H" 1" 2H" 1H" 1H" 5" 2" 1H" 3" 2" 2" 6" 2H" 2" 3H" 2H" 2H" 7" 3" 2H" 4" 3" 3" 8" 3H" 3" 4H" 3H" 3H" 9" 4" 3H" 5" 4" 4" 10" 4H" 4" 5H" 4H" 4H" 11" 5" 4H" 6" 5" 5" 12" 5H" 5" 6H" 5H" 5H" 13" 6" 5H" 7" 6" 6" 14" 6H" 6" 7H" 6H" 6H" 15" 7" 6H" 8" 7" 7" 16" 7H" 7" 8H" 7H" 7H" 17" 8" 7H" 9" 8" 8" 18" 8H" 8" 9H" 8H" 8H" 19" 9" 8H" 10" 9" 9" 20" 9H" 9" 10H" 9H" ® 1955 Diamond Street San Marcos, CA 92078 800 777-4852 • www.quiltinaday.com Making Eight Half Triangle Squares 1. Cut one square of Background and one square of Triangle fabric. 2. Draw diagonal lines on the wrong side of Background square. 3. Place marked Background square right sides together with Triangle square. Pin. 4. Set up machine with thread that shows against the wrong side of both fabrics. It is important to see the stitching. 5. Sew exactly ¼” from lines with 15 stitches to the inch or a 2.0 on computerized macjine. 6. Press to set seams. 7. Lay on cutting mat grid lines. 8. Without moving fabric, cut squares horizontally and vertically. 9. Cut on both diagonal lines. There should be a total of 8 closed triangles.
    [Show full text]