Geometric Constructions

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Presentation transcript:

Geometric Constructions Unit 2 – Congruence 2/23/2019 Algebra 1 Institute

History Thales (600 BC) Hippocrates (440 BC) Euclid (300 BC) In Golden Age of Greece, Greek mathematicians make a game of geometric constructions, using compass and straight edge alone. Over the years, people have tried to find what they could and could not construct with only a compass and straight edge. Euclid (300 BC) Archimedes (220 BC) 2/23/2019

History Egyptian (1800 BC) Babylonian (3000 BC) Chinese (400BC) Indian Ancient Egyptians used compass to mark off distance. Over the years, people have tried to find what they could and could not construct with only a compass and straight edge. Indian (900 BC) 2/23/2019

How to make Geometric figures Sketching/Drawing By free hand Or any tools Put some pictures to illustrate Compass-and-straight-edge Construction 2/23/2019

Warm up Construct the following using pencil, compass and straight edge only. Do not fold the paper! 3/4 unit ( 1 unit is given below) Hints_1: Hints_2: Bisect a given angle 2/23/2019

Share with others Do you see different ways of construction? 3/4 unit Bisect an angle Hints_1: Hints_2: 2/23/2019

Rules of Construction Euclid’s construction rule Construction rules that are well accepted by modern society 2/23/2019

What are the advantages of applying Euclid’s Rule? Open Discussion What are the advantages of applying Euclid’s Rule? When we use less tools to build geometric figures, we apply more geometric properties. The compass method is usually more precise as it does not rely on the correct measurement of angles or lengths. 2/23/2019

Can we construct the following ? Equilateral Triangle Isosceles Triangle 2/23/2019

Find the center of a circle If it is on a piece of paper…? What if it is a steel ring…? Can we copy a given triangle? What if it is a steel disk…? 2/23/2019

Inscribed Regular Polygons An inscribed equilateral hexagon 2/23/2019

Inscribed Regular Polygons An inscribed equilateral triangle 2/23/2019

Inscribed Regular Polygons An inscribed Square 2/23/2019

Inscribed Regular n-gons Inscribed regular dodecagon 2/23/2019

Over Lapping The minor Arc CK= 1/3 of the circumference The minor arc CF=1/4 of the Circumference The minor arc FK = 1/3 - 1/4 = 1/12 of the circumference 2/23/2019

Can we find a geometric length of any number using the unit of 1 ? 2 units ¾ unit …? 1/3 5/4 0.4 2/23/2019

Construction of Fractions n=a/b 2/23/2019

Irrational Numbers Hint_1: Hint_2: 2/23/2019

Square root of any integer “n” Beautiful? Tedious? 2/23/2019

What if n is a large number? Applying geometric mean 2/23/2019

Beauty of Geometric Construction 2/23/2019

Modern architectures and arts 2/23/2019