PYTHAGORAS THEOREM
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Pythagoras' theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It introduces Pythagorean triples, where the sides of a right triangle have integer values that follow this relationship. The proof of the theorem shows that the areas of squares constructed on the sides of a right triangle follow the same relationship, demonstrating why the hypotenuse must be the longest side. The theorem is only applicable to right triangles.
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mathsproject-
1. The document discusses properties and congruence criteria of triangles. It defines triangles as having three sides, three angles, and three vertices.
2. There are five criteria for determining if two triangles are congruent: side-angle-side (SAS), angle-side-angle (ASA), angle-angle-side (AAS), side-side-side (SSS), and right-angle-hypotenuse-side (RHS).
3. Additional properties discussed include: angles opposite equal sides are equal; sides opposite equal angles are equal; the sum of any two sides is greater than the third side.
triangles ppt.pptx
1. The document discusses properties and congruence criteria of triangles. It defines triangles as having three sides, three angles, and three vertices.
2. There are five criteria for determining if two triangles are congruent: side-angle-side (SAS), angle-side-angle (ASA), angle-angle-side (AAS), side-side-side (SSS), and right-angle-hypotenuse-side (RHS).
3. Additional properties discussed include: angles opposite equal sides are equal; sides opposite equal angles are equal; the sum of any two sides is greater than the third side.
Mensuration notes and_ solved problems
- The document discusses concepts related to mensuration including Pythagoras theorem, sine rule, cosine rule, areas and perimeters of various shapes, volumes, and surface areas.
- Formulas are provided for calculating lengths, areas, perimeters, volumes, and surface areas of shapes like triangles, rectangles, circles, cylinders, spheres, cones, and prisms.
- Examples are given to demonstrate how to use the formulas and break down irregular shapes into regular components to find measurements.
Pythagorean Theorem and its various Proofs
The document discusses several proofs of the Pythagorean theorem provided by different mathematicians. It begins by stating the theorem, then provides 6 different proofs: the first given by President James Garfield in 1876 using a trapezoid approach; the second using similarity of triangles; the third constructing a square from 4 copies of a right triangle; the fourth also using similarity; the fifth by rearrangement of the formula; and the sixth using a geometric representation of the areas. It also discusses some applications of the theorem in fields like architecture, navigation, and coordinate geometry.
Realiabilty and item analysis in assessment
This document discusses polygons and their classifications. It begins by defining what a polygon is - a closed plane figure formed by connecting three or more line segments at their endpoints. It then discusses different types of polygons including regular vs irregular, convex vs concave, simple vs complex, and names polygons based on the number of sides. Specific polygon types like triangles, quadrilaterals, and properties such as interior angles, area, and perimeter are also covered. Formulas to calculate area, sum of interior angles, and measure of central angle are provided.
TRIANGLES
Triangles are geometric shapes with three sides and three angles. They can be categorized based on their angles as right, obtuse, or acute triangles and based on their sides as equilateral, isosceles, or scalene triangles. Key properties of triangles include the angle sum property that the interior angles sum to 180 degrees, Pythagorean theorem relating the sides of a right triangle, and congruence rules to determine if two triangles are identical in shape and size. Triangles are fundamental building blocks that are important across many fields including engineering, trigonometry, and studying distant objects.
Pythagorean theorem
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It is named after Pythagoras, who is often credited with its proof, although it was known by earlier mathematicians as well. There are many different proofs of the theorem, including Pythagoras' original proof by rearrangement of triangles and Euclid's algebraic proof in his Elements which constructs squares on each side and uses their areas to prove the relationship.
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PYTHAGORAS THEOREM
- 1. Name : SARAVANAN P Dept no: 22-EDM-16 Level 2 TOPIC:PYTHAGORAS THEOREM
- 2. INTRODUCTION INTRODUCTION Pythagoras theorem is basically used to find the length of an unknown side and the angle of a triangle. By this theorem, we can derive the base,perpendicular & hypotenuse formulas.
- 3. INTRODUCTION INTRODUCTION pythagoras Theorem (also called Pythagorean Theorem) is an important topic in Mathematics, which explains the relation between the sides of a right-angled triangle. The sides of the right triangle are also called Pythagorean triples.
- 4. STATEMENT pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“
- 5. The sides of this triangle have been named Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple.
- 7. FORMULA FORMULA Consider the triangle given above: Where “a” is the perpendicular, “b” is the base, “c” is the hypotenuse. According to the definition, the Pythagoras Theorem formula is given as: " Hypotenuse² = Perpendicular² + Base² c² = a² + b² "
- 10. PROOF PROOF The side opposite to the right angle (90°) is the longest side (known as Hypotenuse) because the side opposite to the greatest angle is the longest.
- 12. PROOF PROOF Consider three squares of sides a, b, c mounted on the three sides of a triangle having the same sides as shown. By Pythagoras Theorem – Area of square “a” + Area of square “b” = Area of square “c”
- 13. PROOF PROOF We know, △ADB ~ △aBC therefore, ADAB= ABAC(corresponding sides of similar triangle) Or, AB² = AD ×AC……………………………..……..(1) Also, △BDC ~△ABC therefore,
- 14. CD/BC=BC/AC OR, BC²= CD × AC ……………………………………..(2) Adding the equations (1) and (2) we get, AB²+ BC² = AD × AC + CD × AC AB² + BC²= AC (AD + CD) Since, AD + CD = AC Therefore, AC² = AB² + BC² hence, The Pythagoras Theorem is proved
- 15. EXAMPLE
- 16. NOTE NOTE PYTHAGOREAN THEOREM IS ONLY APPLICABLE TO RIGHT-ANGLED TRIANGLE.