The proofs of the Early Greeks 2800 B.C. – 450 B.C.

Slides:

Advertisements

Similar presentations

1.3 Use Midpoint and Distance Formulas

Advertisements

Pythagoras Pythagoras was a Greek scholar and philosopher in the late century BC. Known as “the father of numbers, his teachings covered a variety of areas.

7/3/ : Midsegments of Triangles 3.7: Midsegments of Triangles and Trapezoids G1.2.2: Construct and justify arguments and solve multistep problems.

The point halfway between the endpoints of a line segment is called the midpoint. A midpoint divides a line segment into two equal parts.

The Distance and Midpoint Formula

Pythagorean Triples Big Idea:.

The Pythagorean Theorem and Its Converse

Euclid’s Classification of Pythagorean triples Integer solutions to x 2 +y 2 =z 2. These give right triangles with integer sides. Examples (2n+1,2n 2 +2n,2n.

9.2 The Pythagorean Theorem Geometry Mrs. Spitz Spring 2005.

Chapter 2: Euclid’s Proof of the Pythagorean Theorem

© by S-Squared, Inc. All Rights Reserved. 1.Find the distance, d, and the midpoint, m, between (4, − 2) and (2, 6) Distance Formula: d = (x.

12.3 The Pythagorean Theorem CORD Mrs. Spitz Spring 2007.

PrasadVarious Numbers1 VEDIC MATHEMATICS : Various Numbers T. K. Prasad

MCHS ACT Review Plane Geometry. Created by Pam Callahan Spring 2013 Edition.

THE DISTANCE FORMULA ALGEBRA 1 CP. WARM UP Can the set of numbers represent the lengths of the sides of a right triangle? 4, 5, 6.

Holt CA Course Triangles Warm Up Warm Up California Standards California Standards Lesson Presentation Lesson PresentationPreview.

Section 5.1 Polynomials Addition And Subtraction.

1.3 Distance and Midpoints

Benchmark 40 I can find the missing side of a right triangle using the Pythagorean Theorem.

Section 11.6 Pythagorean Theorem. Pythagorean Theorem: In any right triangle, the square of the length of the hypotenuse equals the sum of the squares.

Section 7.2 – The Quadratic Formula. The solutions to are The Quadratic Formula

Section 8-1: The Pythagorean Theorem and its Converse.

9.2 The Pythagorean Theorem Geometry Mrs. Gibson Spring 2011.

Introduction Triangles are typically thought of as simplistic shapes constructed of three angles and three segments. As we continue to explore this shape,

Euclid zVery little is known about his life zProfessor at the university of Alexandria z“There is no royal road to geometry”

Holt CA Course Triangles Vocabulary Triangle Sum Theoremacute triangle right triangleobtuse triangle equilateral triangle isosceles triangle scalene.

Geometry Section 7.2 Use the Converse of the Pythagorean Theorem.

Triangle Sum Theorem In a triangle, the three angles always add to 180°: A + B + C = 180° 38° + 85° + C = 180° C = 180° C = 57°

Pythagorean Theorem - Thurs, Oct 7

4.7 – Square Roots and The Pythagorean Theorem Day 2.

9.2 The Pythagorean Theorem

Chapter 9 Review Square RootsTriangles The Pythagorean Theorem Real Numbers Distance and Midpoint Formulas

Quadratic Equations and Problem Solving. The square of a number minus twice the number is sixty three.

Section 8-3 The Converse of the Pythagorean Theorem.

Exploring. Pythagorean Theorem For any right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the.

Pythagoras Theorem Proof of the Pythagorean Theorem using Algebra.

The proofs of the Early Greeks 2800 B.C. – 450 B.C.

5.5 Triangle Inequality. Objectives: Use the Triangle Inequality.

Lesson 35: Special Right Triangles and the Pythagorean Theorem

The Right Triangle and The Pythagorean Theorem

Right Triangle The sides that form the right angle are called the legs. The side opposite the right angle is called the hypotenuse.

9.2 The Pythagorean Theorem

The Converse of the Pythagorean Theorem

Chapter 7 Proofs and Conditional Probability

The Addition Postulates and some important definitions, Module 1

9.2 The Pythagorean Theorem

9-2 Pythagorean Theorem.

1-6 Midpoint & Distance in the Coordinate Plane

Pythagorean Theorem RIGHT TRIANGLE Proof that the formula works!

Direct Proof and Counterexample I

The Pythagorean Theorem c a b.

Parallel lines and Triangles Intro Vocabulary

PROVING THE PYTHAGOREAN THEOREM

Find which two of the segments XY, ZY, and ZW are congruent.

Warm Up Take out your placemat and discuss it with your neighbor.

Objectives/Assignment

10.3 and 10.4 Pythagorean Theorem

Pythagorean Theorem a²+ b²=c².

11.7 and 11.8 Pythagorean Thm..

Warm Up Take out your placemat and discuss it with your neighbor.

Introduction Triangles are typically thought of as simplistic shapes constructed of three angles and three segments. As we continue to explore this shape,

9.2 The Pythagorean Theorem

Area and Perimeter Review

If a triangle is a RIGHT TRIANGLE, then a2 + b2 = c2.

The Pythagorean Theorem

Examples of Mathematical Proof

Pythagorean Theorem and it’s converse

The Pythagoras Theorem c a a2 + b2 = c2 b.

Presentation transcript:

The proofs of the Early Greeks 2800 B.C. – 450 B.C.

Pythagorean Theorem  Write an equation illustrating the relationship between the area of the rectangles, the area of the white smaller square, and the area of the large square  Write an equation illustrating the relationship between the smaller squares, the rectangles and the larger square.  Use transitivity and the subtraction property of equality.

Pythagorean formula for integer solutions Exercise 1: Find a right triangle with side length equal to 9. Exercise 2: Find a right triangle with side length 14.

Irrational Numbers! Relatively prime = no common factors

Irrational Numbers!

Greek’s Idea of Algebra Euclid’s Proposition 4 of Book II: If a straight line divided into any two parts, the square on the whole line is equal to the sum of the squares on the two parts together with twice the rectangle contained by the two parts. How does this mean that? 

Translating Euclid  Square on the line: A square whose side length is equal to the length of the given segment Square ABDE  Rectangle contained by the two parts: The rectangle whose side lengths are given by the two parts Rectangle ACDF  A straight line is divided: Plot a point somewhere on the line segment. Point C

Proposition 5 of Book II If a straight line is divided equally and also unequally, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. Let us translate

Proposition 5 of Book II If a straight line is divided equally and also unequally, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. Create a straight line segment AB Divide it equally : Plot the midpoint. Call this M. And also unequally: Plot a non- midpoint point, Q, on AB. Construct the rectangle contained by the unequal parts. Construct the square on the points M and Q. Add these areas up. The sum should be equal to the square on AM.

Does your shape look like this? Verify the postulate!

Method of Proportions Find the relationship between x, a, b, and c

Method of Proportions