DOCSLIB.ORG

7 Quadrilaterals and Other Polygons

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
7 Quadrilaterals and Other Polygons 7 Quadrilaterals and Other Polygons 7.1 Angles of Polygons 7.2 Properties of Parallelograms 7.3 Proving That a Quadrilateral Is a Parallelogram 7.4 Properties of Special Parallelograms 7.5 Properties of Trapezoids and Kites SEE the Big Idea DiamondDiamond (p.((p. 406)406) WiWindow d ((p. 39395)5) AmAmusementusement Park Ride Ride (p(p. 377) ArArrowrow (p(p.. 37373)3) GazeboGazebo (p.(p. 36365)5) hhs_geo_pe_07co.indds_geo_pe_07co.indd 356356 11/19/15/19/15 111:411:41 AAMM MMaintainingaintaining MathematicalMathematical ProficiencyProficiency Using Structure to Solve a Multi-Step Equation Example 1 Solve 3(2 + x) = −9 by interpreting the expression 2 + x as a single quantity. 3(2 + x) = −9 Write the equation. 3(2 + x) −9 — = — Divide each side by 3. 3 3 2 + x = −3 Simplify. −2 −2 Subtract 2 from each side. x = −5 Simplify. Solve the equation by interpreting the expression in parentheses as a single quantity. 1. 4(7 − x) = 16 2. 7(1 − x) + 2 = −19 3. 3(x − 5) + 8(x − 5) = 22 Identifying Parallel and Perpendicular Lines Example 2 Determine which of the lines are parallel and which are perpendicular. a b y c Find the slope of each line. (−4, 3) d 3 − (−3) (2, 2) Line a: m = — = −3 2 (3, 2) −4 − (−2) −1 − (−4) Line b: m = — = −3 −2 (1, −1) 4 x 1 − 2 (−4, 0) (4, −2) − − −2 = —2 ( 2) = − Line c: m − 4 3 4 (−2, −3) (2, −4) −4 2 − 0 1 Line d: m = — = — 2 − (−4) 3 Because lines a and b have the same slope, lines a and b are parallel. Because 1 — − = − 3 ( 3) 1, lines a and d are perpendicular and lines b and d are perpendicular. Determine which lines are parallel and which are perpendicular. 4. a y d 5. c y b 6. (−4, 4) b y c 4 4 a 4 (3, 4) (−2, 4) (0, 3) (−4, 2) (−2, 2) d 2 d 2 (−2, 2) (4, −2) (0, 1) (3, 1) c (−3, 0) (1, 0) (2, 1) (3, 1) − x x − x 4 − − 1 3 4 2 4 b ( 3, 3) −2 −2 (−3, −2) −2 (2, −3) (−3, −2) (0, −4) (0, −3) (−2, −2) a (3, −3) (4, −4) (4, −4) 7. ABSTRACT REASONING Explain why interpreting an expression as a single quantity does not contradict the order of operations. Dynamic Solutions available at BigIdeasMath.com 357 hhs_geo_pe_07co.indds_geo_pe_07co.indd 357357 11/19/15/19/15 111:411:41 AAMM MMathematicalathematical PPracticesractices Mathematically profi cient students use diagrams to show relationships. Mapping Relationships CCoreore CConceptoncept Classifi cations of Quadrilaterals quadrilaterals parallelograms trapezoids rhombusessquares rectangles kites Writing Statements about Quadrilaterals Use the Venn diagram above to write three true statements about different types of quadrilaterals. SOLUTION Here are three true statements that can be made about the relationships shown in the Venn diagram. • All rhombuses are parallelograms. • Some rhombuses are rectangles. • No trapezoids are parallelograms. MMonitoringonitoring PProgressrogress Use the Venn diagram above to decide whether each statement is true or false. Explain your reasoning. 1. Some trapezoids are kites. 2. No kites are parallelograms. 3. All parallelograms are rectangles. 4. Some quadrilaterals are squares. 5. Example 1 lists three true statements based on the Venn diagram above. Write six more true statements based on the Venn diagram. 6. A cyclic quadrilateral is a quadrilateral that can be circumscribed by a circle so that the circle touches each vertex. Redraw the Venn diagram so that it includes cyclic quadrilaterals. 358 Chapter 7 Quadrilaterals and Other Polygons hhs_geo_pe_07co.indds_geo_pe_07co.indd 358358 11/19/15/19/15 111:411:41 AAMM 7.1 Angles of Polygons EEssentialssential QQuestionuestion What is the sum of the measures of the interior angles of a polygon? The Sum of the Angle Measures of a Polygon Work with a partner. Use dynamic geometry software. a. Draw a quadrilateral and a pentagon. Find the sum of the measures of the interior angles of each polygon. Sample B G F A C H E I D b. Draw other polygons and fi nd the sums of the measures of their interior angles. Record your results in the table below. CONSTRUCTING VIABLE ARGUMENTS Number of sides, n 3 456789 S To be profi cient in math, Sum of angle measures, you need to reason c. Plot the data from your table in a coordinate plane. inductively about data. d. Write a function that fi ts the data. Explain what the function represents. Measure of One Angle in a Regular Polygon Work with a partner. a. Use the function you found in Exploration 1 to write a new function that gives the measure of one interior angle in a regular polygon with n sides. b. Use the function in part (a) to fi nd the measure of one interior angle of a regular pentagon. Use dynamic geometry software to check your result by constructing a regular pentagon and fi nding the measure of one of its interior angles. c. Copy your table from Exploration 1 and add a row for the measure of one interior angle in a regular polygon with n sides. Complete the table. Use dynamic geometry software to check your results. CCommunicateommunicate YourYour AnswerAnswer 3. What is the sum of the measures of the interior angles of a polygon? 4. Find the measure of one interior angle in a regular dodecagon (a polygon with 12 sides). Section 7.1 Angles of Polygons 359 hhs_geo_pe_0701.indds_geo_pe_0701.indd 359359 11/19/15/19/15 111:501:50 AAMM 7.1 Lesson WWhathat YYouou WWillill LLearnearn Use the interior angle measures of polygons. Core VocabularyVocabulary Use the exterior angle measures of polygons. diagonal, p. 360 Using Interior Angle Measures of Polygons equilateral polygon, p. 361 p. 361 In a polygon, two vertices that are endpoints of Polygon ABCDE equiangular polygon, the same side are called consecutive vertices. p. 361 C regular polygon, A diagonal of a polygon is a segment that Previous joins two nonconsecutive vertices. B D polygon As you can see, the diagonals from one vertex convex divide a polygon into triangles. Dividing a diagonals interior angles polygon with n sides into (n − 2) triangles A E exterior angles shows that the sum of the measures of the A and B are consecutive vertices.— — interior angles of a polygon is a multiple Vertex B has two diagonals, BD and BE. of 180°. TTheoremheorem Theorem 7.1 Polygon Interior Angles Theorem The sum of the measures of the interior angles n n − ° 2 REMEMBER of a convex -gon is ( 2) ⋅ 180 . 3 1 m∠ + m∠ + . + m∠n = n − ° A polygon is convex when 1 2 ( 2) ⋅ 180 4 no line that contains a side 6 5 of the polygon contains Proof Ex. 42 (for pentagons), p. 365 n = 6 a point in the interior of the polygon. Finding the Sum of Angle Measures in a Polygon Find the sum of the measures of the interior angles of the fi gure. SOLUTION The fi gure is a convex octagon. It has 8 sides. Use the Polygon Interior Angles Theorem. (n − 2) ⋅ 180° = (8 − 2) ⋅ 180° Substitute 8 for n. = 6 ⋅ 180° Subtract. = 1080° Multiply. The sum of the measures of the interior angles of the fi gure is 1080°. MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com 1. The coin shown is in the shape of an 11-gon. Find the sum of the measures of the interior angles. 360 Chapter 7 Quadrilaterals and Other Polygons hhs_geo_pe_0701.indds_geo_pe_0701.indd 360360 11/19/15/19/15 111:501:50 AAMM Finding the Number of Sides of a Polygon The sum of the measures of the interior angles of a convex polygon is 900°. Classify the polygon by the number of sides. SOLUTION Use the Polygon Interior Angles Theorem to write an equation involving the number of sides n. Then solve the equation to fi nd the number of sides. (n − 2) ⋅ 180° = 900° Polygon Interior Angles Theorem n − 2 = 5 Divide each side by 180°. n = 7 Add 2 to each side. The polygon has 7 sides. It is a heptagon. CCorollaryorollary Corollary 7.1 Corollary to the Polygon Interior Angles Theorem The sum of the measures of the interior angles of a quadrilateral is 360°. Proof Ex. 43, p. 366 Finding an Unknown Interior Angle Measure x 108° 121° Find the value of in the diagram. SOLUTION x° 59° The polygon is a quadrilateral. Use the Corollary to the Polygon Interior Angles Theorem to write an equation involving x. Then solve the equation. x° + 108° + 121° + 59° = 360° Corollary to the Polygon Interior Angles Theorem x + 288 = 360 Combine like terms. x = 72 Subtract 288 from each side. The value of x is 72. MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com 2. The sum of the measures of the interior angles of a convex polygon is 1440°. Classify the polygon by the number of sides. 3. The measures of the interior angles of a quadrilateral are x°, 3x°, 5x°, and 7x°. Find the measures of all the interior angles. In an equilateral polygon, In an equiangular A regular polygon is all sides are congruent. polygon, all angles in the a convex polygon that interior of the polygon are is both equilateral and congruent. equiangular. Section 7.1 Angles of Polygons 361 hhs_geo_pe_0701.indds_geo_pe_0701.indd 361361 11/19/15/19/15 111:501:50 AAMM Finding Angle Measures in Polygons A home plate for a baseball fi eld is shown. AB a. Is the polygon regular? Explain your reasoning.
Load more

Recommended publications
  • 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.
    [Show full text]
  • ON a PROOF of the TREE CONJECTURE for TRIANGLE TILING BILLIARDS Olga Paris-Romaskevich
    ON A PROOF OF THE TREE CONJECTURE FOR TRIANGLE TILING BILLIARDS Olga Paris-Romaskevich To cite this version: Olga Paris-Romaskevich. ON A PROOF OF THE TREE CONJECTURE FOR TRIANGLE TILING BILLIARDS. 2019. hal-02169195v1 HAL Id: hal-02169195 https://hal.archives-ouvertes.fr/hal-02169195v1 Preprint submitted on 30 Jun 2019 (v1), last revised 8 Oct 2019 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ON A PROOF OF THE TREE CONJECTURE FOR TRIANGLE TILING BILLIARDS. OLGA PARIS-ROMASKEVICH To Manya and Katya, to the moments we shared around mathematics on one winter day in Moscow. Abstract. Tiling billiards model a movement of light in heterogeneous medium consisting of homogeneous cells in which the coefficient of refraction between two cells is equal to −1. The dynamics of such billiards depends strongly on the form of an underlying tiling. In this work we consider periodic tilings by triangles (and cyclic quadrilaterals), and define natural foliations associated to tiling billiards in these tilings. By studying these foliations we manage to prove the Tree Conjecture for triangle tiling billiards that was stated in the work by Baird-Smith, Davis, Fromm and Iyer, as well as its generalization that we call Density property.
    [Show full text]
  • Circle and Cyclic Quadrilaterals
    Circle and Cyclic Quadrilaterals MARIUS GHERGU School of Mathematics and Statistics University College Dublin 33 Basic Facts About Circles A central angle is an angle whose vertex is at the center of ² the circle. It measure is equal the measure of the intercepted arc. An angle whose vertex lies on the circle and legs intersect the ² cirlc is called inscribed in the circle. Its measure equals half length of the subtended arc of the circle. AOC= contral angle, AOC ÙAC \ \ Æ ABC=inscribed angle, ABC ÙAC \ \ Æ 2 A line that has exactly one common point with a circle is ² called tangent to the circle. 44 The tangent at a point A on a circle of is perpendicular to ² the diameter passing through A. OA AB ? Through a point A outside of a circle, exactly two tangent ² lines can be drawn. The two tangent segments drawn from an exterior point to a cricle are equal. OA OB, OBC OAC 90o ¢OAB ¢OBC Æ \ Æ \ Æ Æ) ´ 55 The value of the angle between chord AB and the tangent ² line to the circle that passes through A equals half the length of the arc AB. AC,BC chords CP tangent Æ Æ ÙAC ÙAC ABC , ACP \ Æ 2 \ Æ 2 The line passing through the centres of two tangent circles ² also contains their tangent point. 66 Cyclic Quadrilaterals A convex quadrilateral is called cyclic if its vertices lie on a ² circle. A convex quadrilateral is cyclic if and only if one of the fol- ² lowing equivalent conditions hold: (1) The sum of two opposite angles is 180o; (2) One angle formed by two consecutive sides of the quadri- lateral equal the external angle formed by the other two sides of the quadrilateral; (3) The angle between one side and a diagonal equals the angle between the opposite side and the other diagonal.
    [Show full text]
  • Angles in a Circle and Cyclic Quadrilateral MODULE - 3 Geometry
    Angles in a Circle and Cyclic Quadrilateral MODULE - 3 Geometry 16 Notes ANGLES IN A CIRCLE AND CYCLIC QUADRILATERAL You must have measured the angles between two straight lines. Let us now study the angles made by arcs and chords in a circle and a cyclic quadrilateral. OBJECTIVES After studying this lesson, you will be able to • verify that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle; • prove that angles in the same segment of a circle are equal; • cite examples of concyclic points; • define cyclic quadrilterals; • prove that sum of the opposite angles of a cyclic quadrilateral is 180o; • use properties of cyclic qudrilateral; • solve problems based on Theorems (proved) and solve other numerical problems based on verified properties; • use results of other theorems in solving problems. EXPECTED BACKGROUND KNOWLEDGE • Angles of a triangle • Arc, chord and circumference of a circle • Quadrilateral and its types Mathematics Secondary Course 395 MODULE - 3 Angles in a Circle and Cyclic Quadrilateral Geometry 16.1 ANGLES IN A CIRCLE Central Angle. The angle made at the centre of a circle by the radii at the end points of an arc (or Notes a chord) is called the central angle or angle subtended by an arc (or chord) at the centre. In Fig. 16.1, ∠POQ is the central angle made by arc PRQ. Fig. 16.1 The length of an arc is closely associated with the central angle subtended by the arc. Let us define the “degree measure” of an arc in terms of the central angle.
    [Show full text]
  • Cyclic Quadrilaterals — the Big Picture Yufei Zhao Yufeiz@Mit.Edu
    Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao Cyclic Quadrilaterals | The Big Picture Yufei Zhao yufeiz@mit.edu 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]
  • Cyclic and Bicentric Quadrilaterals G
    Cyclic and Bicentric Quadrilaterals G. T. Springer Email: gt.springer@hp.com Hewlett-Packard Calculators and Educational Software Abstract. In this hands-on workshop, participants will use the HP Prime graphing calculator and its dynamic geometry app to explore some of the many properties of cyclic and bicentric quadrilaterals. The workshop will start with a brief introduction to the HP Prime and an overview of its features to get novice participants oriented. Participants will then use ready-to-hand constructions of cyclic and bicentric quadrilaterals to explore. Part 1: Cyclic Quadrilaterals The instructor will send you an HP Prime app called CyclicQuad for this part of the activity. A cyclic quadrilateral is a convex quadrilateral that has a circumscribed circle. 1. Press ! to open the App Library and select the CyclicQuad app. The construction consists DEGH, a cyclic quadrilateral circumscribed by circle A. 2. Tap and drag any of the points D, E, G, or H to change the quadrilateral. Which of the following can DEGH never be? • Square • Rhombus (non-square) • Rectangle (non-square) • Parallelogram (non-rhombus) • Isosceles trapezoid • Kite Just move the points of the quadrilateral around enough to convince yourself for each one. Notice HDE and HE are both inscribed angles that subtend the entirety of the circle; ≮ ≮ likewise with DHG and DEG. This leads us to a defining characteristic of cyclic ≮ ≮ quadrilaterals. Make a conjecture. A quadrilateral is cyclic if and only if… 3. Make DEGH into a kite, similar to that shown to the right. Tap segment HE and press E to select it. Now use U and D to move the diagonal vertically.
    [Show full text]
  • Fun Problems in Geometry and Beyond
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 097, 21 pages Fun Problems in Geometry and Beyond Boris KHESIN †y and Serge TABACHNIKOV ‡z †y Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada E-mail: khesin@math.toronto.edu URL: http://www.math.toronto.edu/khesin/ ‡z Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA E-mail: tabachni@math.psu.edu URL: http://www.personal.psu.edu/sot2/ Received November 17, 2019; Published online December 11, 2019 https://doi.org/10.3842/SIGMA.2019.097 Abstract. We discuss fun problems, vaguely related to notions and theorems of a course in differential geometry. This paper can be regarded as a weekend “treasure\treasure chest”chest" supplemen- ting the course weekday lecture notes. The problems and solutions are not original, while their relation to the course might be so. Key words: clocks; spot it!; hunters; parking; frames; tangents; algebra; geometry 2010 Mathematics Subject Classification: 00A08; 00A09 To Dmitry Borisovich Fuchs on the occasion of his 80th anniversary Áîëüøå çàäà÷, õîðîøèõ è ðàçíûõ! attributed to Winston Churchill (1918) This is a belated birthday present to Dmitry Borisovich Fuchs, a mathematician young at 1 heart.1 Many of these problems belong to the mathematical folklore. We did not attempt to trace them back to their origins (which might be impossible anyway), and the given references are not necessarily the first mentioning of these problems. 1 Problems 1. AA wall wall clock clock has has the the minute minute and and hour hour hands hands of of equal equal length.
    [Show full text]
  • Areas of Polygons Inscribed in a Circle
    Discrete Comput Geom 12:223-236 (1994) G iii try 1994 Springer-VerlagNew York Inc. Areas of Polygons Inscribed in a Circle D. P. Robbins Center for Communications Research, Institute for Defense Analyses, Princeton, NJ 08540, USA robbins@ ccr-p.ida.org Abstract. Heron of Alexandria showed that the area K of a triangle with sides a, b, and c is given by lr = x/s(s - a)~s - b• - c), where s is the semiperimeter (a + b + c)/2. Brahmagupta gave a generalization to quadrilaterals inscribed in a circle. In this paper we derive formulas giving the areas of a pentagon or hexagon inscribed in a circle in terms of their side lengths. While the pentagon and hexagon formulas are complicated, we show that each can be written in a surprisingly compact form related to the formula for the discriminant of a cubic polynomial in one variable. 1. Introduction Since a triangle is determined by the lengths, a, b, c of its three sides, the area K of the triangle is determined by these three lengths. The well-known formula K = x/s(s - aXs - bXs - c), (1.1) where s is the semiperimeter (a + b + c)/2, makes this dependence explicit. (This formula is usually ascribed to Heron of Alexandria, c. 60 B.c., although some attribute it to Archimedes.) For polygons of more than three sides, the lengths of the sides do not determine the polygon or its area. However, if we impose the condition that the polygon be convex and cyclic (i.e., inscribed in a circle), then the area of the polygon is uniquely determined.
    [Show full text]
  • Brahmagupta's Derivation of the Area of a Cyclic Quadrilateral
    Brahmagupta’s derivation of the area of a cyclic quadrilateral Satyanad Kichenassamy To cite this version: Satyanad Kichenassamy. Brahmagupta’s derivation of the area of a cyclic quadrilateral. Historia Mathematica, Elsevier, 2010, 37 (1), pp.28-61. 10.1016/j.hm.2009.08.004. halshs-00443604 HAL Id: halshs-00443604 https://halshs.archives-ouvertes.fr/halshs-00443604 Submitted on 12 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Historia Mathematica 37 (2010) 28{61 http://dx.doi.org/10.1016/j.hm.2009.08.004 Brahmagupta's derivation of the area of a cyclic quadrilateral Satyanad Kichenassamy1 Abstract. This paper shows that Propositions XII.21{27 of Brahmagupta's Br¯ahma- sphut.asiddh¯anta (628 a.d.) constitute a coherent mathematical discourse leading to the expression of the area of a cyclic quadrilateral in terms of its sides. The radius of the circumcircle is determined by considering two auxiliary quadrilaterals. Observing that a cyclic quadrilateral is split by a diagonal into two triangles with the same circumcenter and the same circumradius, the result follows, using the tools available to Brahmagupta. The expression for the diagonals (XII.28) is a consequence.
    [Show full text]
  • The Quad Squad Conjectures About Quadrilaterals
    2 The Quad Squad Conjectures About Quadrilaterals Warm Up Learning Goals Classify each figure using as • Use diagonals to draw quadrilaterals. many names as possible. • Make conjectures about the diagonals of 1. special quadrilaterals. • Make conjectures about the angle relationships of special quadrilaterals. • Categorize quadrilaterals based upon their properties. • Make conjectures about the midsegments 2. of quadrilaterals. • Understand that the vertices of cyclic quadrilaterals lie on the same circle. 3. Key Terms • coincident • isosceles trapezoid • interior angle of a • midsegment © Carnegie Learning, Inc. 4. polygon • cyclic quadrilateral • kite You have classifi ed quadrilaterals by their side measurements and side relationships. What conjectures can you make about diff erent properties of quadrilaterals? LESSON 2: The Quad Squad • M1-127 GGEO_SE_M01_T02_L02.inddEO_SE_M01_T02_L02.indd 112727 55/22/18/22/18 112:512:51 PPMM GETTING STARTED Cattywampus A quadrilateral may be convex or concave. The quadrilaterals you are most familiar with—trapezoids, parallelograms, rectangles, rhombi, and squares—are convex. For a polygon to be convex it contains all of the line segments connecting any pair of points. It is concave if and only if at least Think one pair of its interior angles is greater than 180°. about: Consider the two quadrilaterals shown. A quadrilateral has exactly two diagonals. Why can a concave quadrilateral have Convex Concave only one angle greater than 180°? 1. Draw the diagonals in the two quadrilaterals shown. What do you notice? 2. Make a conjecture about the diagonals of a convex quadrilateral and about the diagonals of a concave quadrilateral. The diagonals of any convex quadrilateral create two pairs of vertical angles © Carnegie Learning, Inc.
    [Show full text]
  • Bicentric Quadrilateral Central Configurations 1
    BICENTRIC QUADRILATERAL CENTRAL CONFIGURATIONS JAUME LLIBRE1 AND PENGFEI YUAN2 Abstract. A bicentric quadrilateral is a tangential cyclic quadri- lateral. In a tangential quadrilateral the four sides are tangents to an inscribed circle, and in a cyclic quadrilateral the four vertices lie on a circumscribed circle. In this paper we classify all planar cen- tral configurations of the 4-body problem, where the four bodies are at the vertices of a bicentric quadrilateral. 1. Introduction and statement of the results The well-known Newtonian n-body problem concerns with the mo- tion of n mass points with positive mass mi moving under their mutual attraction in Rd in accordance with Newton’s law of gravitation. The equations of the motion of the n-body problem are : n mj(xi xj) x¨i = − , 1 i n, − X r3 ≤ ≤ j=1,j=i ij 6 where we have taken the unit of time in such a way that the Newtonian d gravitational constant be one, and xi R (i = 1,...,n) denotes the ∈ position vector of the i-body, rij = xi xj is the Euclidean distance between the i-body and the j-body.| − | Alternatively the equations of the motion can be written mix¨i = iU(x), 1 i n, ∇ ≤ ≤ where x =(x1,...,xn), and m m U(x)= i j , X x x 1 i<j n i j ≤ ≤ | − | is the potential of the system. 2010 Mathematics Subject Classification. 70F07,70F15. Key words and phrases. Convex central configuration, four-body problem, bi- centric quadrilateral. 1 cat http://www.gsd.uab. 2 J.LLIBREANDP.YUAN The solutions of the 2-body problem (also called the Kepler problem) has been completely solved.
    [Show full text]