PYTHAGORAS THEOREM
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The Pythagoras theorem helps to find the side of the triangle. This is possible only for Right angled triangles and the important note is that every triangle can be splitted into two right angle triangles
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PYTHAGORAS THEOREM.pptx
The document explains Pythagoras theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It provides the definition of a right-angled triangle, the formula for the theorem, and a proof of the theorem. It also discusses applications of the theorem such as determining if a triangle is right-angled or calculating an unknown side of a right triangle.
PYTHAGORAS THEOREM
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.
Chapter 6, triangles For Grade -10
This document provides an overview of triangles, including their properties, types, similarity, areas of similar triangles, and Pythagorean theorem. It defines triangles as polygons with three sides and angles, and notes the angle sum property. Triangles are classified by angle and side length into right, obtuse, acute, isosceles, equilateral, and scalene varieties. Similarity is discussed, along with criteria like equal angles and proportional sides. Areas of similar triangles are directly proportional to a scale factor. Finally, the Pythagorean theorem and its applications to find missing sides of right triangles are covered.
chapter 6, triangles
This document provides an overview of triangles, including their properties, types, similarity, areas of similar triangles, and the Pythagorean theorem. It defines triangles as polygons with three sides and angles, and notes their interior angles sum to 180 degrees. Triangles are classified by angle and side length into right, acute, obtuse, isosceles, equilateral, and scalene varieties. Similarity is discussed, along with criteria like equal angles and proportional sides. Areas of similar triangles are shown to be proportional to a ratio of corresponding sides. Finally, the Pythagorean theorem and its applications to find missing sides of right triangles are covered.
pythagoras theoreem.pptx
The document discusses the history and derivation of the Pythagorean theorem. It states that the theorem was first invented by the Greek mathematician Pythagoras, but was also discovered independently by ancient Indian and Babylonian mathematicians as early as 1900 BC. The document then provides the derivation of the theorem using similar right triangles, and proves the converse is also true. It concludes by discussing applications of the theorem such as determining if a triangle is right-angled or calculating unknown sides.
Plane Figures.pptx
The document discusses different types and properties of triangles. It begins by defining a triangle as a three-sided polygon with three angles and three vertices. It then describes various triangle classifications based on side lengths (scalene, isosceles, equilateral) and angle measures (acute, right, obtuse). Various properties of triangles are outlined, such as angle sum, exterior angles, and relationships between sides and angles. Formulas for calculating perimeter and area of triangles are also provided. The document concludes by presenting theoretical proofs and experimental verifications of several triangle theorems.
Core sub math_att_4pythagoreantheorem
The document provides information about the Pythagorean theorem:
1) It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
2) It gives examples of right triangles that satisfy the theorem, such as ones with sides of 3, 4, 5 or 5, 12, 13.
3) It includes an animated proof of the theorem showing how the area of the square on the hypotenuse equals the combined areas of the squares on the other two sides.
triangles class9.pptx
1. A triangle is a closed figure formed by three intersecting lines, with three sides, three angles, and three vertices.
2. There are five criteria to determine 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. Key properties of triangles 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.
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PYTHAGORAS THEOREM.pptx
The document explains Pythagoras theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It provides the definition of a right-angled triangle, the formula for the theorem, and a proof of the theorem. It also discusses applications of the theorem such as determining if a triangle is right-angled or calculating an unknown side of a right triangle.
PYTHAGORAS THEOREM
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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This document provides an overview of triangles, including their properties, types, similarity, areas of similar triangles, and Pythagorean theorem. It defines triangles as polygons with three sides and angles, and notes the angle sum property. Triangles are classified by angle and side length into right, obtuse, acute, isosceles, equilateral, and scalene varieties. Similarity is discussed, along with criteria like equal angles and proportional sides. Areas of similar triangles are directly proportional to a scale factor. Finally, the Pythagorean theorem and its applications to find missing sides of right triangles are covered.
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This document provides an overview of triangles, including their properties, types, similarity, areas of similar triangles, and the Pythagorean theorem. It defines triangles as polygons with three sides and angles, and notes their interior angles sum to 180 degrees. Triangles are classified by angle and side length into right, acute, obtuse, isosceles, equilateral, and scalene varieties. Similarity is discussed, along with criteria like equal angles and proportional sides. Areas of similar triangles are shown to be proportional to a ratio of corresponding sides. Finally, the Pythagorean theorem and its applications to find missing sides of right triangles are covered.
pythagoras theoreem.pptx
The document discusses the history and derivation of the Pythagorean theorem. It states that the theorem was first invented by the Greek mathematician Pythagoras, but was also discovered independently by ancient Indian and Babylonian mathematicians as early as 1900 BC. The document then provides the derivation of the theorem using similar right triangles, and proves the converse is also true. It concludes by discussing applications of the theorem such as determining if a triangle is right-angled or calculating unknown sides.
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The document discusses different types and properties of triangles. It begins by defining a triangle as a three-sided polygon with three angles and three vertices. It then describes various triangle classifications based on side lengths (scalene, isosceles, equilateral) and angle measures (acute, right, obtuse). Various properties of triangles are outlined, such as angle sum, exterior angles, and relationships between sides and angles. Formulas for calculating perimeter and area of triangles are also provided. The document concludes by presenting theoretical proofs and experimental verifications of several triangle theorems.
Core sub math_att_4pythagoreantheorem
The document provides information about the Pythagorean theorem:
1) It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
2) It gives examples of right triangles that satisfy the theorem, such as ones with sides of 3, 4, 5 or 5, 12, 13.
3) It includes an animated proof of the theorem showing how the area of the square on the hypotenuse equals the combined areas of the squares on the other two sides.
triangles class9.pptx
1. A triangle is a closed figure formed by three intersecting lines, with three sides, three angles, and three vertices.
2. There are five criteria to determine 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. Key properties of triangles 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.
Trigonometry part 1
This document discusses trigonometry and trigonometric ratios. It defines trigonometry as the study of relationships between the sides and angles of a triangle. It then defines the trigonometric ratios of sine, cosine, and tangent for a right triangle in terms of the sides adjacent to, opposite to, and hypotenuse of an angle. It provides examples of using trigonometric ratios to solve for missing sides or angles of a right triangle. The document also discusses trigonometric ratios of complementary angles and provides the specific trigonometric ratios for 30, 60, 45, and 90 degree angles.
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PYTHAGOREAN THEOREM
This PPT prepared by Teacher Educator from Thiagarajar College of Preceptors Madurai for B.Ed Trainees.
Pythagorean Theorem
Pythagoras was an ancient Greek thinker, but he was not the founder of the Pythagorean theorem. That honor goes to his followers, known as the Pythagorean Brotherhood, who established the theorem over 100 years after Pythagoras' death. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This relationship has many practical applications in fields like engineering, construction, physics, and astronomy that involve calculating distances.
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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.
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This document discusses various metric relationships and theorems related to triangles and circles. It covers concepts like congruency, similarity, the Pythagorean theorem, right triangle area formulas, products of sides, altitudes, projections, proportional means, 30 and 45 degree theorems, angle bisectors, and sample exam questions. Key relationships discussed include the Pythagorean theorem, products of sides equaling products of hypotenuse and altitude, and various theorems relating side lengths based on angles or segments in right triangles.
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- A quadrilateral is a four-sided polygon. Quadrilaterals can be simple or complex, and simple ones can be convex or concave. The interior angles of any simple quadrilateral sum to 360 degrees.
- A circle is the set of all points in a plane equid
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The document provides an explanation of the Pythagorean theorem using examples of right triangles found in baseball diamonds and ladders. It begins by defining a right triangle and the key terms of hypotenuse, legs, and the Pythagorean theorem formula a2 + b2 = c2. It then works through two word problems, solving for missing side lengths by setting up the appropriate equations and calculations based on the given information.
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The document provides an explanation of the Pythagorean theorem using examples of right triangles found in baseball diamonds and ladders. It begins by defining a right triangle and its components - the hypotenuse and two legs. It then states the Pythagorean theorem formula that the sum of the squares of the two legs equals the square of the hypotenuse. Several word problems are worked through step-by-step using the theorem to calculate missing side lengths of right triangles.
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The document discusses Pythagoras' theorem, which 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 provides examples of applying the theorem to calculate missing side lengths. Pythagoras discovered this relationship around 2500 years ago. The theorem can be used to solve problems involving right triangles, such as calculating the total distance traveled by someone who walks two legs of a right triangle.
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1. The document defines triangles and their properties including three sides, three angles, and three vertices.
2. It explains 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).
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1) A triangle is a three-sided polygon with three vertices and three edges.
2) Triangles can be classified based on side lengths (equilateral, isosceles, scalene) or interior angles (right, acute, obtuse).
3) The interior angles of any triangle always sum to 180 degrees. Congruent triangles have the same shape and size, while similar triangles have the same angle measures but sides proportional in length.
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This document covers several metric relationships and theorems related to triangles and right triangles. It discusses concepts like congruency, similarity, angle bisector theorem, Pythagorean theorem, products of sides, geometric mean, altitude to hypotenuse theorem, projections of sides, proportional mean theorem, 30 degree theorem, median theorem, and includes example problems to solve involving right triangles.
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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.
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PYTHAGORAS THEOREM
- 1. PYTHAGORAS THEOREM Your best friend to solve Right-angled triangles. Presented by Vaishika G
- 2. Overview › Pythagoras Theorem (also called Pythagorean Theorem) is an important topic in Mathematics, which explains the relation between the sides of a right-angled triangle. › 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 and hypotenuse formulas. › The sides of the right triangle are also called Pythagorean triples.
- 3. RIGHT – ANGLED TRIANGLE A right-angled triangle is a type of triangle that has one of its angles equal to 90 degrees. The sides that include the right angle are perpendicular and the base of the triangle. The third side is called the hypotenuse, which is the longest side of all three sides.
- 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. Formula Consider the triangle given in the previous slide: Where “a” is the perpendicular, “b” is the base, “c” is the hypotenuse. According to the definition, the Pythagoras Theorem formula is given as: 2
- 6. PROOF Given: A right-angled triangle ABC, right-angled at B. To Prove: AC2 = AB2 + BC2 Construction: Draw a perpendicular BD meeting AC at D.
- 7. Proof › We know, △ADB ~ △ABC › Therefore, › 𝐴𝐷 𝐴𝐵 = 𝐴𝐵 𝐴𝐶 (corresponding sides of similar triangles) › Or, AB2 = AD × AC ……………………………..……..(1) › Also, △BDC ~△ABC › Therefore, › 𝐶𝐷 𝐵𝐶 = 𝐵𝐶 𝐴𝐶
- 8. Proof › 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 Pythagorean theorem is proved.
- 9. Application • To know if the triangle is a right-angled triangle or not. • In a right-angled triangle, we can calculate the length of any side if the other two sides are given. • To find the diagonal of a square.
- 10. Wrapping it up The Pythagorean Theorem can be used only on _____triangles. When should the Pythagorean Theorem be used? What should be done first when solving a word problem involving the Pythagorean Theorem? What must be done before writing the answer to a Pythagorean Theorem problem? Right When the length of 2 sides are known and the length of 3rd side is needed Draw and label triangle Check to see whether the answer should be rounded or not