Definition: a Quadrilateral Is a Polygon with 4 Sides. a Diagonal of a Quadrilat- Eral Is a Line Segment Whose End-Points Are Op
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How Can You Put Shapes Together to Make Other Shapes?
Lesson 27 Understand Tiling in Rectangles Name: How can you put shapes together to make other shapes? Study the example showing how to put shapes together to make other shapes. Then solve Problems 1–4. Example How can you use smaller shapes to make a circle? Look at the shapes in the green box Draw lines to show how you could use the shapes Write how many of each shape you would need 2 4 1 Draw a line to show how you could use squares like the one in the green box to make a rectangle How many squares would you need? squares 2 Draw lines to show how you could use triangles like the one in the green box to make a rectangle How many triangles would you need? triangles ©Curriculum Associates, LLC Copying is not permitted. Lesson 27 Understand Tiling in Rectangles 283 Solve. 3 Draw lines to show how you could use hexagons to make the first shape below Then show how you could use trapezoids to make the second shape How many of each shape would you need? hexagons trapezoids 4 Draw lines to show how you could use triangles to make the first shape below Then show how you could use trapezoids to make the second shape Then show how you could use rhombuses to make the third shape How many of each shape would you need? triangles trapezoids rhombuses 284 Lesson 27 Understand Tiling in Rectangles ©Curriculum Associates, LLC Copying is not permitted. Lesson 27 Name: Draw and Count Squares Study the example showing how to draw and count squares. -
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. -
Quadrilateral Theorems
Quadrilateral Theorems Properties of Quadrilaterals: If a quadrilateral is a TRAPEZOID then, 1. at least one pair of opposite sides are parallel(bases) If a quadrilateral is an ISOSCELES TRAPEZOID then, 1. At least one pair of opposite sides are parallel (bases) 2. the non-parallel sides are congruent 3. both pairs of base angles are congruent 4. diagonals are congruent If a quadrilateral is a PARALLELOGRAM then, 1. opposite sides are congruent 2. opposite sides are parallel 3. opposite angles are congruent 4. consecutive angles are supplementary 5. the diagonals bisect each other If a quadrilateral is a RECTANGLE then, 1. All properties of Parallelogram PLUS 2. All the angles are right angles 3. The diagonals are congruent If a quadrilateral is a RHOMBUS then, 1. All properties of Parallelogram PLUS 2. the diagonals bisect the vertices 3. the diagonals are perpendicular to each other 4. all four sides are congruent If a quadrilateral is a SQUARE then, 1. All properties of Parallelogram PLUS 2. All properties of Rhombus PLUS 3. All properties of Rectangle Proving a Trapezoid: If a QUADRILATERAL has at least one pair of parallel sides, then it is a trapezoid. Proving an Isosceles Trapezoid: 1st prove it’s a TRAPEZOID If a TRAPEZOID has ____(insert choice from below) ______then it is an isosceles trapezoid. 1. congruent non-parallel sides 2. congruent diagonals 3. congruent base angles Proving a Parallelogram: If a quadrilateral has ____(insert choice from below) ______then it is a parallelogram. 1. both pairs of opposite sides parallel 2. both pairs of opposite sides ≅ 3. -
Cyclic Quadrilateral: Cyclic Quadrilateral Theorem and Properties of Cyclic Quadrilateral Theorem (For CBSE, ICSE, IAS, NET, NRA 2022)
9/22/2021 Cyclic Quadrilateral: Cyclic Quadrilateral Theorem and Properties of Cyclic Quadrilateral Theorem- FlexiPrep FlexiPrep Cyclic Quadrilateral: Cyclic Quadrilateral Theorem and Properties of Cyclic Quadrilateral Theorem (For CBSE, ICSE, IAS, NET, NRA 2022) Get unlimited access to the best preparation resource for competitive exams : get questions, notes, tests, video lectures and more- for all subjects of your exam. A quadrilateral is a 4-sided polygon bounded by 4 finite line segments. The word ‘quadrilateral’ is composed of two Latin words, Quadric meaning ‘four’ and latus meaning ‘side’ . It is a two-dimensional figure having four sides (or edges) and four vertices. A circle is the locus of all points in a plane which are equidistant from a fixed point. If all the four vertices of a quadrilateral ABCD lie on the circumference of the circle, then ABCD is a cyclic quadrilateral. In other words, if any four points on the circumference of a circle are joined, they form vertices of a cyclic quadrilateral. It can be visualized as a quadrilateral which is inscribed in a circle, i.e.. all four vertices of the quadrilateral lie on the circumference of the circle. What is a Cyclic Quadrilateral? In the figure given below, the quadrilateral ABCD is cyclic. ©FlexiPrep. Report ©violations @https://tips.fbi.gov/ 1 of 5 9/22/2021 Cyclic Quadrilateral: Cyclic Quadrilateral Theorem and Properties of Cyclic Quadrilateral Theorem- FlexiPrep Let us do an activity. Take a circle and choose any 4 points on the circumference of the circle. Join these points to form a quadrilateral. Now measure the angles formed at the vertices of the cyclic quadrilateral. -
Lesson 3: Rectangles Inscribed in Circles
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY Lesson 3: Rectangles Inscribed in Circles Student Outcomes . Inscribe a rectangle in a circle. Understand the symmetries of inscribed rectangles across a diameter. Lesson Notes Have students use a compass and straightedge to locate the center of the circle provided. If necessary, remind students of their work in Module 1 on constructing a perpendicular to a segment and of their work in Lesson 1 in this module on Thales’ theorem. Standards addressed with this lesson are G-C.A.2 and G-C.A.3. Students should be made aware that figures are not drawn to scale. Classwork Scaffolding: Opening Exercise (9 minutes) Display steps to construct a perpendicular line at a point. Students follow the steps provided and use a compass and straightedge to find the center of a circle. This exercise reminds students about constructions previously . Draw a segment through the studied that are needed in this lesson and later in this module. point, and, using a compass, mark a point equidistant on Opening Exercise each side of the point. Using only a compass and straightedge, find the location of the center of the circle below. Label the endpoints of the Follow the steps provided. segment 퐴 and 퐵. Draw chord 푨푩̅̅̅̅. Draw circle 퐴 with center 퐴 . Construct a chord perpendicular to 푨푩̅̅̅̅ at and radius ̅퐴퐵̅̅̅. endpoint 푩. Draw circle 퐵 with center 퐵 . Mark the point of intersection of the perpendicular chord and the circle as point and radius ̅퐵퐴̅̅̅. 푪. Label the points of intersection . -
The Geometry of Diagonal Groups
The geometry of diagonal groups R. A. Baileya, Peter J. Camerona, Cheryl E. Praegerb, Csaba Schneiderc aUniversity of St Andrews, St Andrews, UK bUniversity of Western Australia, Perth, Australia cUniversidade Federal de Minas Gerais, Belo Horizonte, Brazil Abstract Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan{Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combi- natorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied in general. The main purpose of this paper is to describe and characterise such struct- ures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan{Scott theorem, which are defined over finite characteristically simple groups, our construction works over arbitrary groups, finite or infinite. A diagonal semilattice depends on a dimension m and a group T . For m = 2, it is a Latin square, the Cayley table of T , though in fact any Latin square satisfies our combinatorial axioms. However, for m > 3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in great profusion but higher- dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set Ω, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions turn out to be equivalent for diagonal groups). -
Properties of Equidiagonal Quadrilaterals (2014)
Forum Geometricorum Volume 14 (2014) 129–144. FORUM GEOM ISSN 1534-1178 Properties of Equidiagonal Quadrilaterals Martin Josefsson Abstract. We prove eight necessary and sufficient conditions for a convex quadri- lateral to have congruent diagonals, and one dual connection between equidiag- onal and orthodiagonal quadrilaterals. Quadrilaterals with both congruent and perpendicular diagonals are also discussed, including a proposal for what they may be called and how to calculate their area in several ways. Finally we derive a cubic equation for calculating the lengths of the congruent diagonals. 1. Introduction One class of quadrilaterals that have received little interest in the geometrical literature are the equidiagonal quadrilaterals. They are defined to be quadrilat- erals with congruent diagonals. Three well known special cases of them are the isosceles trapezoid, the rectangle and the square, but there are other as well. Fur- thermore, there exists many equidiagonal quadrilaterals that besides congruent di- agonals have no special properties. Take any convex quadrilateral ABCD and move the vertex D along the line BD into a position D such that AC = BD. Then ABCD is an equidiagonal quadrilateral (see Figure 1). C D D A B Figure 1. An equidiagonal quadrilateral ABCD Before we begin to study equidiagonal quadrilaterals, let us define our notations. In a convex quadrilateral ABCD, the sides are labeled a = AB, b = BC, c = CD and d = DA, and the diagonals are p = AC and q = BD. We use θ for the angle between the diagonals. The line segments connecting the midpoints of opposite sides of a quadrilateral are called the bimedians and are denoted m and n, where m connects the midpoints of the sides a and c. -
Simple Polygons Scribe: Michael Goldwasser
CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons Scribe: Michael Goldwasser Algorithms for triangulating polygons are important tools throughout computa- tional geometry. Many problems involving polygons are simplified by partitioning the complex polygon into triangles, and then working with the individual triangles. The applications of such algorithms are well documented in papers involving visibility, motion planning, and computer graphics. The following notes give an introduction to triangulations and many related definitions and basic lemmas. Most of the definitions are based on a simple polygon, P, containing n edges, and hence n vertices. However, many of the definitions and results can be extended to a general arrangement of n line segments. 1 Diagonals Definition 1. Given a simple polygon, P, a diagonal is a line segment between two non-adjacent vertices that lies entirely within the interior of the polygon. Lemma 2. Every simple polygon with jPj > 3 contains a diagonal. Proof: Consider some vertex v. If v has a diagonal, it’s party time. If not then the only vertices visible from v are its neighbors. Therefore v must see some single edge beyond its neighbors that entirely spans the sector of visibility, and therefore v must be a convex vertex. Now consider the two neighbors of v. Since jPj > 3, these cannot be neighbors of each other, however they must be visible from each other because of the above situation, and thus the segment connecting them is indeed a diagonal. -
Unit 6 Visualising Solid Shapes(Final)
• 3D shapes/objects are those which do not lie completely in a plane. • 3D objects have different views from different positions. • A solid is a polyhedron if it is made up of only polygonal faces, the faces meet at edges which are line segments and the edges meet at a point called vertex. • Euler’s formula for any polyhedron is, F + V – E = 2 Where F stands for number of faces, V for number of vertices and E for number of edges. • Types of polyhedrons: (a) Convex polyhedron A convex polyhedron is one in which all faces make it convex. e.g. (1) (2) (3) (4) 12/04/18 (1) and (2) are convex polyhedrons whereas (3) and (4) are non convex polyhedron. (b) Regular polyhedra or platonic solids: A polyhedron is regular if its faces are congruent regular polygons and the same number of faces meet at each vertex. For example, a cube is a platonic solid because all six of its faces are congruent squares. There are five such solids– tetrahedron, cube, octahedron, dodecahedron and icosahedron. e.g. • A prism is a polyhedron whose bottom and top faces (known as bases) are congruent polygons and faces known as lateral faces are parallelograms (when the side faces are rectangles, the shape is known as right prism). • A pyramid is a polyhedron whose base is a polygon and lateral faces are triangles. • A map depicts the location of a particular object/place in relation to other objects/places. The front, top and side of a figure are shown. Use centimetre cubes to build the figure. -
Teach the Square…And the Rectangle
f e a t u r e Teach the square…and the rectangle eaching these two geometric needs of ancient civilizations. familiar and trusted shapes, they Tshapes is like a line from an (Some believe the square symbol- represent order and stability. old song: “You can’t have one izes civilization.) Ancient peoples Because a square has four equal without the other.” needed to measure plots of land, sides, it can also be said to sym- The square and rectangle go build structures, collect taxes, bolize the seasons (spring, sum- hand in hand—maybe not like describe constellations, and create mer, fall, winter), cardinal direc- “love and marriage” or “horse calendars, all of which required tions (north, east, south, west), and carriage,” but perhaps best numbers, shapes, and calculations. and the elements (fire, water, air, understood in terms of the other. Of the basic geometric shapes in earth). Children have undoubtedly had today’s world, the rectangle is the In nature, we see no naturally experience with the two shapes in most common. The rectangle and formed squares or rectangles. objects all around them. They its square soul-mate form the Trees, branches, and leaves, for begin learning geometric shapes structure of most buildings. As example, are uneven, rounded, in a more formal way in preschool z T and are usually expected to know gae how to make and take apart N shapes by the time they finish kindergarten. by susa oto Texas Child Care has posted two ph previous articles on teaching geo- metric shapes: Go round: Teach the circle shape in Spring 2015 and Teach the triangle in Spring 2017. -
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. -
Quadrilaterals
Name: Quadrilaterals Match the quadrilateral with its definition. a. 1. All sides are the same length. There are four right angles. Parallelogram b. 2. There is only one pair of parallel sides. Rectangle c. 3. Opposite sides are parallel and the same length. There are four right angles. Trapezoid d. 4. There are two pairs of parallel sides. All sides are the same length. Rhombus 5. There are two pairs of opposite parallel e. sides. Square 6. List two ways a rectangle and square are alike and one way in which they are different. Super Teacher Worksheets - www.superteacherworksheets.com ANSWER KEY Quadrilaterals Match the quadrilateral with its definition. a. e 1. All sides are the same length. There are four right angles. Parallelogram b. c 2. There is only one pair of parallel sides. Rectangle c. b 3. Opposite sides are parallel and the same length. There are four right angles. Trapezoid d. d 4. There are two pairs of parallel sides. All sides are the same length. Rhombus a 5. There are two pairs of opposite parallel e. sides. Square 6. List two ways a rectangle and square are alike and one way in which they are different. Rectangles and squares are alike in that both have two pairs of parallel sides and four right angles. They are different because a square must have four equal sides lengths and a rectangle has two pairs of equal side lengths. Super Teacher Worksheets - www.superteacherworksheets.com.