Collection of exercises in ELEMENTARY GEOMETRY for Programme Primary school teachers Leni Lvovskä October 2019 Geometry has two treasures: a pythagorean theorem and a golden section. The first one is worth gold, the second one is more precious stone. Johannes Kepler (1571 - 1630) [10] Introduction This collection of elementary geometry problems was developed as a supporting material to the geometry textbooks for the future elementary school teachers. These texts namely contain only a limited number of exercises and no solved tasks. This booklet offers the students a number of solved tasks as well as another set of exercises. At the same time, it follows the current trend of inter-subject connections and in the provided tasks and examples shows how geometry is related to the other subjects as well as to the world around us. Many tasks work with the magnetic kit Geomag. If you do not have it, these tasks can be demonstrated using skewers and balls of modeling. For the creation of illustrations, the GeoGebra software was used. It is therefore easy to use the GeoGebra tutorial software directly in the classroom or on a standalone task. There are direct references to selected dynamic applets and stepped constructions for specific constructions. This text was developed with the support of the project MUNI / FR / 1193/2018, Innovation of four subjects Geometry for Teachers of the 1st level of the elementary school with Geomag kit and Geogebra educational software at the Faculty of Education, Masaryk University in Brno. Many thanks to Helena Durnová for the preparation of the English version of this text and Pavel Kříž for the support of the typesetting in the I^TfrjXsystem. 5 1 Historical development of geometry Exercises 1.1. Attachment at the end of the text contains a set of pictures. Sort these images into three groups by assigning them to one of the three basic geometric figures: circle, square, equilateral triangle. Discuss the pictures in the groups. Why did you assign these pictures to the groups belonging to a circle, square or equilateral triangle. Are there images that could be assigned to two or even to all three groups? An example of such a discussion: This picture can be seen as a regular hexagon, so it can be assigned to an equilateral triangle, since a regular hexagon consists of six equal equilateral triangles. A regular hexagon is a regular polygon, ie it can be copied and inscribed a circle, so we can assign it to a circle. This image can also be seen as a wireframe cube. If appropriate, explain this view to classmates who can not see it. 6 A regular hexagon or cube? Exercises 1.2. How did geometry begin to form in the distant past? (Formulate the answer in several sentences.) Exercises 1.3. What do you know about the books we call Euclid's Elements! 32 EL EMEN TS EUCLID> Exfkiurd n*i Dinfri ■ I i a ATfil' Ami moJleAJre Mtthod. , with the U S E S of each 'PR 0P0SIT10N In all the P-artSQf tlK (4 ATHE MA-TICKS. JJ/Clamlc Prncrois Jlillitt l}'Clia]«.„ J.-fmtf. Ooiiemnof V R E K C H, Corrected »bd Augmented, awl I]1mJI rated wild Nine Copper Plates, aiul theEffiEioofB$c LID, By to WUliatiy Philomath. t.0 X JJ O A"; Printed forPJjf'/ip/^.Glcbcnnfccr, at the yfl/.u ind HtrctJti in the Pmirrty, near Chujfuk, KSSj. Exercises 1.4. Correctly associate branches of geometry with the names of important mathematicians, who worked in them: 7 • René Descartes (1596 - 1650), • Johann Carl Friedrich Gauss (1777 - 1855), • Georg Friedrich Bernhard Riemann (1826 - 1866), • David Hilbert (1862 - 1943). a) German mathematician and physicist. He was interested in geometry, mathematical analysis, number theory, astronomy, electrostatics, geodesy and optics. He strongly influenced most of these fields of knowledge. He stood at the birth of non-Euclidean geometry. b) His work La Geometrie is often considered the beginning of analytic geometry as a science. c) German mathematician, who in his work Foundations of Geometry constructed discipline currently called Euclidean geometry, he created the system of axioms of Euclidean geometry. d) German mathematician, who contributed significantly to the development of mathematical analysis and differential geometry. Algebraic geometry and complex surface theory were also developed on the basic of his ideas, which became the core of differential geometry on manifolds and topology. Solution: René Descartes (b), Johann Carl Friedrich Gauss (a), Georg Friedrich Bernhard Riemann (d), David Hilbert (c). Exercises 1.5. Explain the difference between an axiom and a mathematical theorem. Give an example of an axiom and a mathematical theorem. 8 2 Basic geometric formations and their properties Example 2.1. Which geometric formations can arise as the intersection of two half-lines that are lie on the same line? Show and describe. Solution: Point, line segment, line, half line. Example 2.2. Investigate all possible relative positions of three different lines lying in one plane. Show and describe. Řešeni: Let's denote the lines a, b, c. Then the following situations can occur: a) all lines are mutually parallel, ie. an6 = 0A6nc = 0, b) two lines are parallel and the third line is parallel to them, e.g. aC)b = (bAanc = XABnC = Y c) all lines are mutually parallel and pass through a single common point, anbnc = P, d) all lines are mutually parallel and intersect at different points. Exercises 2.3. Draw a line AB. On the line AB mark: a) C, so that A is between C and B, b) D, so that B is between A and D, c) a point P that does not lie on the AB line but lies on the AD line. Exercises 2.4. Draw line KL. Select D between KL, and mark: a) R, so that K is between R and L, b) S, so that L is between K and S, c) T, so that S is between L, T. Decide which statement is true: KL, 2) RS fl KL = KL, 9 3) ^ RD f] ST = ®, 4) R E KL, Exercises 2.5. There is a line p and a point A that does not lie on it. Draw: point M that belongs to the i—> pA, b) a point P that lies in both halves defined by p, c) a point N that lies in the half-plane opposite to i—y pA. Exercises 2.6. Three different points A, B, C are given. a) How many line segments, half-lines, and lines are determined by these points? How do these numbers depend on the position of the points given? b) Which point sets can be the intersection of two of these line segments (half-lines, lines)? Show and discuss. Exercises 2.7. Let point R lies between P, Q. From half-lines PR, PQ, RP, RQ, QR, QP choose pairs of half-lines that: a) coincide, b) are opposite, c) one is part of the other, d) their intersection is a line segment. Exercises 2.8. Determine what shapes may arise as an intersection of: line segment and a half-plane, b) a half-line and a half-plane, c) a line and a half-plane. For all cases, consider the situation in a single plane. Show and describe. Exercises 2.9. There are n straight lines in the plane, of which no two intersect and no three meet at the same point. How many significant intersections of these lines are there? Example 2.10. How many different lines are determined by n points that lie in one plane and no three lying on one straight line? 10 Solution: For a single point, the task is meaningless. Let's outline the situation for some finite number of points: for two points there will be one straight line, for three points just three lines, four points will determine six lines, five points will be ten lines, etc. Now we can do the following: from each point we lead a line to (n—1) points, but in this way I count them each line twice. The result is: n{n — 1) 2 ' Exercises 2.11. In the plane there are n lines, two of which intersect and no three of them meet the same point. How many intersections there are? Exercises 2.12. Determine what shapes may arise as an intersection of two half-planes. Consider the situation in a single plane. Exercises 2.13. Select points A, B inside one half-plane, which is determined by the line p. Inside the opposite half-plane, select C, D so that the lines AB and CD are parallel to the line p. On line AB select M, on line CD select N. How must the points M, N be choosen so that the line segment MN contains a point of line p lying between M and iV? Example 2.14. Construct a cuboid of ABCDEFGH (using GeoMag or using skewers and plasticine). A) Determine all incident lines with cuboid edges that are with BC: • parallel, • intersecting, • skew. B) Using the points of the cuboid, you list three planes that form a bundle of planes and write down the intersection of these three planes. Reseni: 11 • parallel: -B- AD, -B- EF, -B- • intersecting: ABC n <-> ABF n <-> ABF = <-> AB. Example 2.15. Construct a regular tetrahedral pyramid ABCDV (using GeoMag kit or skewers and plasticine). A) Determine all straight lines specified by A, B, C, D, V that are: • parallel to BC, • intersecting to BC, • skew to BC. B) Using the pyramid points A, B, C, D, V, give an example of the three planes that make up the bunch of planes and write the intersection of the three planes. Solution: 12 • parallel: -H- AD, • intersecting: o AB, o 5V, O CV, O CD, • skew -H- AV, -H- DV. A bunch of planes is made up of planes -H- ABC, -H- ABV a -H- BCV o ABC n o ABV n o 5CV = o {5}. 13 3 Convex and non-convex set, convex and non-convex angle Exercises 3.1. How can we find out whether a geometrical figure is convex or non-convex? Sort geometric shapes into convex and non-convex: a line segment, line, circle, triangle, quadrilateral, pentagon, circle with hole. Exercises 3.2. Look around and try to see the angles determined by the edges of the board or the edges of the bench, parts of the window frame, but also the angles formed, for example, the legs of the chair and the floor. Mark such angles in the illustration. Exercises 3.3. Draw the lines i—> SC and i—> SD. Mark with a red arc the convex angle ^(AB + BC + CA). (1) c Solution: Point S is the inner point of the triangle ABC, so there are three other triangles for which the triangle inequality holds: for triangle ABS: AS + BS > AB, for triangle ACS: AS + CS> AC, for triangle BCS: BS + CS> BC. Adding the right and left sides of the inequalities we get: 2 • AS + 2 • BS + 2 • CS > AB + BC + AC, (2) thus proving inequality (1). Example 4.7. Prove that for the sum of the centroids ta, tb, tc of triangle ABC, the relation is: 1 -(a + b + c) < ta + tb + tc < a + b + c. (3) Solution: First we prove the inequality 1 -(a + b + c) c, for triangle ACC\. tc + | > 6, for triangle.BC.Bi: £5 + | > a. Adding the right and left sides of the inequalities we get: 1 ta + tb + tc + -(a + b + c) > a + b + c, (5) i.e. 1 ta + tb + tc> -(a + b + c). (6) Let us prove the inequality ta + tb + tc< a + b + c. (7) Let the points Ai, Bi, C\ be again the centers of the sides BC, AC and AB of the triangle. Let's construct A' so that A\ is the center line of AA'. The quadrilateral ABAC is a parallelogram, its diagonals halve each other. So AC = BA' holds. The triangular inequality for triangle ABA' implies: 2ta » m (■) 2 s ^ The construction step by step: https://www.geogebra.org/m/u7e5f3qn#material/yzcd6acd 28 Example 4.21. Construct a triangle ABC if a,va,Vb are given. km « 10/10 h. m ® 2 s ^ The construction step by step: https://www.geogebra.org/m/u7e5f3qn#material/rkdepcxf Example 4.22. Construct a triangle ABC if b, c, tc are given. C The construction step by step The record of the construction: The construction step by step: https://www.geogebra.org/m/u7e5f3qn#material/gnr4vvnn 29 Example 4.23. Construct a triangle ABC if b,^y,vc are given. The construction step by step The construction step by step: https://www.geogebra.org/m/u7e5f3qn#material/rssprtnv 30 Example 4.24. Construct a triangle ABC if 7, i>a, i>b are given. \p \ The construction step by step \ c / \ \ /\^X^ \>—\ *y—' Xb The record of the construction: 1) qCX/ = vfc the radius of the circle k2 = vb 5) B; B e <-> q fl CY 6) A ABC HM +, 7/7 WH II The construction step by step: https://www.geogebra.org/m/u7e5f3qn#material/sn3wvaed 31 Example 4.25. Construct a triangle ABC if a, va, b are given. The construction step by step 0 The record of the construction: 1) BC;/BC/ = a 2) w p, w p'; /«pBC/ = /«p'BC/ = va 3) kr k1 (C, b) 4) A; A E k1 n n p 5) A ABC k1 p x \y I B \ 1 p' /c\ J H4 « 9/9 » M- ^_____ r n ® 12 |s '-^ The construction step by step: https://www.geogebra.org/m/u7e5f3qn#material/ntwfvxns 32 Example 4.26. Construct a triangle ABC if a, c, tc are given. The Construction Step by Step The record ofthe construction: 1) AB;/AB/ = c 2) A BAX; MB AX/ = o 3) Sc; Sc e AB A /ASC/ = /SCB/ The construction step by step: https://www.geogebra.org/m/u7e5f3qn#material/trhbazkf 33 Example 4.27. Construct a triangle ABC if 7, va, vc are given. The Construction Step by Step The record of the construction: The construction step by step: https://www.geogebra.org/m/u7e5f3qn#material/njnjbvh9 84 Example 4.28. Construct a triangle ABC if a, 0, rv, are given, where rv is the radius of the circle inscribed to triangle ABC. The construction step by step: https://www.geogebra.org/m/u7e5f3qn#material/w547a5au Exercises 4.29. Construct a triangle ABC if the following are given: a) a + b, j,va b) a — b, 7, c c) a + b + c, a, (3 d) a,b,a — [3 e) a + b + c, a,vc Exercises 4.30. The line AB is given. a) Construct the set of all vertices of the convex angle -QACB = 7, whose arms pass through the endpoints of the line segment AB. b) Construct AABC if \AB\ = 6, 7 = 60°, vc = 4. Exercises 4.31. Construct a triangle ABC if ta,tb,tc are given. 35 Example 4.32. Prove the theorem about the centroids of the triangle: The centroids of each triangle intersect at one point, called the centroid of the triangle, the center of gravity divides each center of gravity into two lines, the one containing the vertex of the triangle is twice the other. Proof: The triangle ABC is given, the points Ai, Bi, C\ are the centers of its sides BC, AC and AB, the lines AAi, BB\ and CC\ are its center of gravity. In this triangle, we consider the AA\ and BB\ lines that intersect at T. We prove that CC\ goes through T. Let's construct a line CT and a point U on it so that the point T is the center of the line CU, i.e. CT = TU. In the triangle AUC, the line B{T is the middle bar and therefore B{T || AU. Since the points Bi, T, B lie on one straight line, it is i BT || AU. Analogously in the BUC triangle, the line A\T is the middle rung and therefore A\T || BU, and hence AT || BU. Hence the quadrilateral ATBU has two parallel sides parallel, i.e. it is a parallelogram and its diagonals AB and TU are bisected. Hence, the center of the AB side, Ci, lies on the line CT. This proves that CC\ goes through T. Therefore, the centroids of the triangle ABC intersect at one point. This point always belongs to the inside of the given triangle. The properties of the middle rungs B\T and A\T of the triangles AUC and BUC and the properties of the parallelogram AUBT further imply: for triangle AUC: B{T = \AU", AU = BT, i.e.. B{T = \BT', 36 for triangle BUC: A{T = \BU, BU ^ AT, i.e. A{T = \AT. This proves that the center of gravity T divides each of the AAi, BB\ lines into two parts, the one containing the vertex of the triangle is twice as long as the other. By repeating the considerations in choosing another pair of lines, we obtain further relation, which imply the correctness of the statement of the second part of the theorem. Exercises 4.33. Prove that two triangles are identical when they match in two sides and in the center of gravity to one of them. Hint: Prove the identity of triangles by using triangles, which are created by dividing the given triangle by the centroid. Exercises 4.34. Above the sides of the acute triangle ABC there are equilateral triangles ABH and ACK. Prove line segments CH and BK are equal. Hint: The assertion follows from the equality of triangles ACH and AKB. Exercises 4.35. The triangle ABC is given. Through its peaks, lines parallel with the opposite sides are drawn. Prove that the intersections of these lines determine the triangle, which is a union of four triangles identical to the triangle ABC. Hint: Use the theorems of triangles identity and properties of pairs of angles between parallel lines. Exercises 4.36. The largest side of the convex quadrilateral ABCD is AB, the smallest CD. Prove that