Florentin Smarandache Compiled and Solved Problems in Geometry and Trigonometry 255 Compiled and Solved Problems in Geometry and Trigonometry FLORENTIN SMARANDACHE 255 Compiled and Solved Problems in Geometry and Trigonometry (from Romanian Textbooks) Educational Publisher 2015 1 Florentin Smarandache Peer reviewers: Prof. Rajesh Singh, School of Statistics, DAVV, Indore (M.P.), India. Dr. Linfan Mao, Academy of Mathematics and Systems, Chinese Academy of Sciences, Beijing 100190, P. R. China. Mumtaz Ali, Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000, Pakistan Prof. Stefan Vladutescu, University of Craiova, Romania. Said Broumi, University of Hassan II Mohammedia, Hay El Baraka Ben M'sik, Casablanca B. P. 7951, Morocco. E-publishing, Translation & Editing: Dana Petras, Nikos Vasiliou AdSumus Scientific and Cultural Society, Cantemir 13, Oradea, Romania Copyright: Florentin Smarandache 1998-2015 Educational Publisher, Columbus, USA ISBN: 978-1-59973-299-2 2 255 Compiled and Solved Problems in Geometry and Trigonometry Table of Content Explanatory Note ..................................................................................................................................................... 4 Problems in Geometry (9th grade) ................................................................................................................... 5 Solutions ............................................................................................................................................................... 11 Problems in Geometry and Trigonometry ................................................................................................. 38 Solutions ............................................................................................................................................................... 42 Other Problems in Geometry and Trigonometry (10th grade) .......................................................... 60 Solutions ............................................................................................................................................................... 67 Various Problems ................................................................................................................................................... 96 Solutions ............................................................................................................................................................... 99 Problems in Spatial Geometry ...................................................................................................................... 108 Solutions ............................................................................................................................................................ 114 Lines and Planes ................................................................................................................................................. 140 Solutions ............................................................................................................................................................ 143 Projections ............................................................................................................................................................. 155 Solutions ............................................................................................................................................................ 159 Review Problems ................................................................................................................................................. 174 Solutions ............................................................................................................................................................ 182 3 Florentin Smarandache Explanatory Note This book is a translation from Romanian of "Probleme Compilate şi Rezolvate de Geometrie şi Trigonometrie" (University of Kishinev Press, Kishinev, 169 p., 1998), and includes problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks for 9th and 10th grade students, in the period 1981-1988, when I was a professor of mathematics at the "Petrache Poenaru" National College in Balcesti, Valcea (Romania), Lycée Sidi El Hassan Lyoussi in Sefrou (Morocco), then at the "Nicolae Balcescu" National College in Craiova and Dragotesti General School (Romania), but also I did intensive private tutoring for students preparing their university entrance examination. After that, I have escaped in Turkey in September 1988 and lived in a political refugee camp in Istanbul and Ankara, and in March 1990 I immigrated to United States. The degree of difficulties of the problems is from easy and medium to hard. The solutions of the problems are at the end of each chapter. One can navigate back and forth from the text of the problem to its solution using bookmarks. The book is especially a didactical material for the mathematical students and instructors. The Author 4 255 Compiled and Solved Problems in Geometry and Trigonometry Problems in Geometry (9th grade) 1. The measure of a regular polygon’s interior angle is four times bigger than the measure of its external angle. How many sides does the polygon have? Solution to Problem 1 2. How many sides does a convex polygon have if all its external angles are obtuse? Solution to Problem 2 3. Show that in a convex quadrilateral the bisector of two consecutive angles forms an angle whose measure is equal to half the sum of the measures of the other two angles. Solution to Problem 3 4. Show that the surface of a convex pentagon can be decomposed into two quadrilateral surfaces. Solution to Problem 4 5. What is the minimum number of quadrilateral surfaces in which a convex polygon with 9, 10, 11 vertices can be decomposed? Solution to Problem 5 6. If (𝐴̂𝐵𝐶) ≡ (𝐴̂′𝐵′𝐶′), then ∃ bijective function 𝑓 = (𝐴̂𝐵𝐶) → (𝐴̂′𝐵′𝐶′) such that for ∀ 2 points 𝑃,𝑄 ∈ (𝐴̂𝐵𝐶), ‖𝑃𝑄‖ = ‖𝑓(𝑃)‖,‖𝑓(𝑄)‖, and vice versa. Solution to Problem 6 5 Florentin Smarandache 7. If ∆𝐴𝐵𝐶 ≡ ∆𝐴′𝐵′𝐶′ then ∃ bijective function 𝑓 = 𝐴𝐵𝐶 → 𝐴′𝐵′𝐶′ such that (∀) 2 points 𝑃,𝑄 ∈ 𝐴𝐵𝐶, ‖𝑃𝑄‖ = ‖𝑓(𝑃)‖,‖𝑓(𝑄)‖, and vice versa. Solution to Problem 7 8. Show that if ∆𝐴𝐵𝐶~∆𝐴′𝐵′𝐶′, then [𝐴𝐵𝐶]~[𝐴′𝐵′𝐶′]. Solution to Problem 8 9. Show that any two rays are congruent sets. The same property for lines. Solution to Problem 9 10. Show that two disks with the same radius are congruent sets. Solution to Problem 10 11. If the function 𝑓:𝑀 → 𝑀′ is isometric, then the inverse function 𝑓−1:𝑀 → 𝑀′ is as well isometric. Solution to Problem 11 12. If the convex polygons 𝐿 = 𝑃 ,𝑃 ,…,𝑃 and 𝐿′ = 𝑃′,𝑃′,…,𝑃′ have |𝑃,𝑃 | ≡ 1 2 𝑛 1 2 𝑛 𝑖 𝑖+1 |𝑃′,𝑃′ | for 𝑖 = 1,2,…,𝑛−1, and 𝑃𝑃̂𝑃 ≡ 𝑃′𝑃′̂𝑃′ , (∀) 𝑖 = 1,2,…,𝑛− 𝑖 𝑖+1 𝑖 𝑖+1 𝑖+2 𝑖 𝑖+1 𝑖+2 2, then 𝐿 ≡ 𝐿′ and [𝐿] ≡ [𝐿′]. Solution to Problem 12 13. Prove that the ratio of the perimeters of two similar polygons is equal to their similarity ratio. Solution to Problem 13 14. The parallelogram 𝐴𝐵𝐶𝐷 has ‖𝐴𝐵‖ = 6, ‖𝐴𝐶‖ = 7 and 𝑑(𝐴𝐶) = 2. Find 𝑑(𝐷,𝐴𝐵). Solution to Problem 14 6 255 Compiled and Solved Problems in Geometry and Trigonometry 15. Of triangles 𝐴𝐵𝐶 with ‖𝐵𝐶‖ = 𝑎 and ‖𝐶𝐴‖ = 𝑏, 𝑎 and 𝑏 being given numbers, find a triangle with maximum area. Solution to Problem 15 16. Consider a square 𝐴𝐵𝐶𝐷 and points 𝐸,𝐹,𝐺,𝐻,𝐼,𝐾,𝐿,𝑀 that divide each side in three congruent segments. Show that 𝑃𝑄𝑅𝑆 is a square and its area is 2 equal to 𝜎[𝐴𝐵𝐶𝐷]. 9 Solution to Problem 16 17. The diagonals of the trapezoid 𝐴𝐵𝐶𝐷 (𝐴𝐵||𝐷𝐶) cut at 𝑂. a. Show that the triangles 𝐴𝑂𝐷 and 𝐵𝑂𝐶 have the same area; b. The parallel through 𝑂 to 𝐴𝐵 cuts 𝐴𝐷 and 𝐵𝐶 in 𝑀 and 𝑁. Show that ||𝑀𝑂|| = ||𝑂𝑁||. Solution to Problem 17 18. 𝐸 being the midpoint of the non-parallel side [𝐴𝐷] of the trapezoid 𝐴𝐵𝐶𝐷, show that 𝜎[𝐴𝐵𝐶𝐷] = 2𝜎[𝐵𝐶𝐸]. Solution to Problem 18 19. There are given an angle (𝐵̂𝐴𝐶) and a point 𝐷 inside the angle. A line through 𝐷 cuts the sides of the angle in 𝑀 and 𝑁. Determine the line 𝑀𝑁 such that the area ∆𝐴𝑀𝑁 to be minimal. Solution to Problem 19 20. Construct a point 𝑃 inside the triangle 𝐴𝐵𝐶, such that the triangles 𝑃𝐴𝐵, 𝑃𝐵𝐶, 𝑃𝐶𝐴 have equal areas. Solution to Problem 20 21. Decompose a triangular surface in three surfaces with the same area by parallels to one side of the triangle. Solution to Problem 21 7 Florentin Smarandache 22. Solve the analogous problem for a trapezoid. Solution to Problem 22 23. We extend the radii drawn to the peaks of an equilateral triangle inscribed in a circle 𝐿(𝑂,𝑟), until the intersection with the circle passing through the peaks of a square circumscribed to the circle 𝐿(𝑂,𝑟). Show that the points thus obtained are the peaks of a triangle with the same area as the hexagon inscribed in 𝐿(𝑂,𝑟). Solution to Problem 23 24. Prove the leg theorem with the help of areas. Solution to Problem 24 25. Consider an equilateral ∆𝐴𝐵𝐶 with ‖𝐴𝐵‖ = 2𝑎. The area of the shaded surface determined by circles 𝐿(𝐴,𝑎),𝐿(𝐵,𝑎),𝐿(𝐴,3𝑎) is equal to the area of the circle sector determined by the minor arc (̂𝐸𝐹) of the circle 𝐿(𝐶,𝑎). Solution to Problem 25 26. Show that the area of the annulus between circles 𝐿(𝑂,𝑟 ) and 𝐿(𝑂,𝑟 ) is 2 2 equal to the area of a disk having as diameter the tangent segment to circle 𝐿(𝑂,𝑟 ) with endpoints on the circle 𝐿(𝑂,𝑟 ). 1 2 Solution to Problem 26 27. Let [𝑂𝐴],[𝑂𝐵] two ⊥ radii of a circle centered at [𝑂]. Take the points 𝐶 and 𝐷 on the minor arc 𝐴̂𝐵𝐹 such that 𝐴̂𝐶≡𝐵̂𝐷 and let 𝐸,𝐹 be the projections of 𝐶𝐷 onto 𝑂𝐵. Show that the area of the surface bounded by [𝐷𝐹],[𝐹𝐸[𝐸𝐶]] and arc 𝐶̂𝐷 is equal to the area of the sector determined by arc 𝐶̂𝐷 of the circle 𝐶(𝑂,‖𝑂𝐴‖). Solution to Problem 27 8 255 Compiled and Solved Problems in Geometry and Trigonometry 28. Find the area of the regular octagon inscribed in a circle of radius 𝑟. Solution to Problem 28 29. Using areas, show that the sum of the distances of a variable point inside the equilateral triangle 𝐴𝐵𝐶 to its sides is constant. Solution to Problem 29 30. Consider a given triangle 𝐴𝐵𝐶 and a variable point 𝑀 ∈ |𝐵𝐶|. Prove that between the distances 𝑥 = 𝑑(𝑀,𝐴𝐵) and 𝑦 = 𝑑(𝑀,𝐴𝐶) is a relation of 𝑘𝑥 + 𝑙𝑦 = 1 type, where 𝑘 and 𝑙 are constant. Solution to Problem 30 31. Let 𝑀 and 𝑁 be the midpoints of sides [𝐵𝐶] and [𝐴𝐷] of the convex quadrilateral 𝐴𝐵𝐶𝐷 and {𝑃} = 𝐴𝑀∩𝐵𝑁 and {𝑄} = 𝐶𝑁∩𝑁𝐷. Prove that the area of the quadrilateral 𝑃𝑀𝑄𝑁 is equal to the sum of the areas of triangles 𝐴𝐵𝑃 and 𝐶𝐷𝑄. Solution to Problem 31 32. Construct a triangle having the same area as a given pentagon. Solution to Problem 32 33. Construct a line that divides a convex quadrilateral surface in two parts with equal areas. Solution to Problem 33 34. In a square of side 𝑙, the middle of each side is connected with the ends of the opposite side. Find the area of the interior convex octagon formed in this way. Solution to Problem 34 9