Mathematics Lecture#1 PDF

Arab Academy for Science, Technology and Maritime Transport Collage of Engineering and Technology– Smart Village Basic and Applied Sciences Department - Cairo Mathematics Lecture#1 Euclidean Geometry: The Five Axioms of Euclidean Geometry -...

Arab Academy for Science, Technology and Maritime Transport Collage of Engineering and Technology– Smart Village Basic and Applied Sciences Department - Cairo Mathematics Lecture#1 Euclidean Geometry: The Five Axioms of Euclidean Geometry - Ruler and Compass Constructions Dr. Mohamed Ibrahim Dr. Mohamed Salah  Introduction These ruins, sometimes buried under desert sand, sometimes covered by jungle vegetation, and sometimes inundated by rising seas and oceans, are silent witnesses of the existence of ancient civilizations with superior practical knowledge of rules of geometry The sphinx and the great The temple of Quetzalcoatl, pyramid, both of also known as the step undetermined antiquity pyramid of Chichen Itza. 2  Introduction A visual proof of the Pythagorean theorem 𝑐 2 = 𝑎2 + 𝑏 2 is given in illustrated Steps 1 to 6 below. STEP 1. We want to show that the area of the largest (blue) square is the sum of the areas of the smaller squares. STEP 2. Split the largest square into the rectangles R and T. These rectangles, as we will see, have the same areas as the squares S and U, respectively. 3  Introduction STEP 3. Shear the rectangles R and T along the line AB. Shears do not affect areas. STEP 4. Then, rotate R around O through 90° as indicated… 4  Introduction STEP 5. … and shear again along the central vertical line. We have shown that R and S have the same area. STEP 6. Finally, perform similar operations to show that T and U are also of the same area. 5  Introduction STEP 5. … and shear again along the central vertical line. We have shown that R and S have the same area. STEP 6. Finally, perform similar operations to show that T and U are also of the same area. 6  Example 1 Express the length of the longest diagonals of the cube in terms of the length of the edges of the cube. 7 Exercises Page 20 8 The Five Axioms of Euclidean Geometry Euclid around 300 B.C. collected all known work in the field of mathematics and arranged it in his famous treatise called Elements. Euclid assumed certain properties, which were not to be proved. These assumptions are actually “obvious universal truths”. 9 The Five Axioms of Euclidean Geometry The five of Euclid’s axioms; are given below : 1) For every point P and every point Q not equal to P, there is a unique line passing through P and Q 2) For every segment AB and for every segment CD, there exists a unique point E such that B is between A and E, and the segment CD is congruent to the segment BE(This axiom tells us that every segment AB can be extended by any length whatsoever) 10 The Five Axioms of Euclidean Geometry 3) For every point O and every point A not equal to O, there exists a circle with the center at O and with radius OA 4) All right angles are congruent. 5) For every line 𝑙 and every point P that does not lie on 𝑙, there exists a unique line 𝑚 through P and parallel to 11 The Five Axioms of Euclidean Geometry A few of the theorems of Euclidean geometry will be needed Theorem 1. The sum of the interior angles of a triangle is always 180° Theorem 2. In the setup of Figure α = β if and only if the lines 𝑙 and m are parallel. 12 The Five Axioms of Euclidean Geometry Theorem 3. If two triangles have equal corresponding angles, then their corresponding sides are proportional. The converse is also true. 𝑎 𝑏 𝑐 𝑎 ′ = ′ = ′ 𝑏 𝑐 Theorem 4 (Thales). A triangle inscribed in a circle so that one of its sides is a diameter of the circle must be a right-angled triangle. 13  Example 2 Construct an equilateral triangle over a given side (this yields a construction of an angle of 60°). 14  Ruler and Compass Constructions CONSTRUCTION 1. Bisecting a Line Segment and Constructing a Perpendicular STEP 1. Given a line segment AB. STEP 3. Find the points of intersection of the circles and draw a line through them. STEP 2. Draw any two intersecting circles of equal radii and centered at the points A and B. 15  Ruler and Compass Constructions CONSTRUCTION 2. Constructing a Line Through a Given Point and Perpendicular to a Given Line STEP 1 Draw a circle centered at the given point C large enough to intersect the line l at two points, denoted by A and B. Then proceed exactly as in Construction 1, constructing the line bisecting the line segment AB. 16  Example 3 Given a line 𝑙 , and a point C out of 𝑙 , construct a parallelogram with one corner at C, and the other two corners on 𝑙. (This will give another construction of the line through C and parallel to 𝑙.) 17  Ruler and Compass Constructions CONSTRUCTION 3. Duplicating an Angle STEP 1. Given an angle α, a line 𝑙, and a point A on the line STEP 2. Draw a circle centered at O and of any radius. Draw the circle centered at A with the same radius. Identify the intersection points B, C, and D. STEP 3. Construct the circle centered at D and with radius equal to BC. Find the intersection point E and join A to E. 18  Ruler and Compass Constructions CONSTRUCTION 4. Drawing a Parallel Line STEP 1. Given a line 𝑙 and a point A out of the line. STEP 2. Draw any line through A intersecting l and identify the angle α. STEP 3. Duplicate the angle α at A as shown (Construction 3), so that the line constructed in the previous step is one of its two sides. 19  Example 4 Double a given angle. 20  Ruler and Compass Constructions CONSTRUCTION 5. Identifying the Center of a Partially Given Circle STEP 1. Given three points A, B, and C, not on the same line. STEP 2. Bisect the line segments AB and BC (see Construction 1). The intersection point O of the bisecting lines is the center of the circle. The radius is any of OA = OB = OC. 21  Ruler and Compass Constructions CONSTRUCTION 6. Constructing a Circle Centered at a Given Point and Touching Another Circle at a Single Point STEP 1. First identify the center O of the given circle. See Construction 5 above. STEP 2. Then join the points A and O and find the intersection point B: this is where the two circles should touch. STEP 3. Finally, draw the circle centered at A and with radius exactly equal to AB. 22  Ruler and Compass Constructions CONSTRUCTION 7. Subdividing a Segment into Equal Parts STEP 1. We are given a line segment AB. STEP 2. Draw any line through A not overlapping with AB. Then mark points A1, A2, A3, and A4 using a compass opened at the same fixed radius (so that all the small segments are equal in length). STEP 3. Connect A4 with B, then construct lines parallel to A4B and through A3, A2, and A1, respectively (as in Construction 4). The intersection points of these three lines with AB give the sought-after points of subdivision. 23  Example 5 Use parallelograms to subdivide a given line segment into five equal parts. 24 Exercises Page 32 25