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The document presents two simple proofs of the Pythagorean theorem given by the 15th century Indian mathematician Ganesh Daiwaidnya. It provides biographical details of Ganesh, noting that he was an astronomer and mathematician born in 1507 AD in Maharashtra, India. The document then shows Ganesh's first proof which uses similar triangles and the second proof which uses trigonometric identities. It concludes by mentioning some Pythagorean triplets invented by Ganesh.

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Mathematicians

The document summarizes the contributions of several important early mathematicians, including Pythagoras who discovered the Pythagorean theorem, Aristotle who studied mathematics in relation to science, and Euclid who wrote The Elements which forms the basis of geometry today. It also discusses mathematicians from other cultures like Al-Khwarizmi from Baghdad who wrote the first book on algebra, as well as Western mathematicians such as Galileo, De L'hopital, Euler, and others who made advances in calculus, trigonometry, and mathematical analysis.

5 mathematicans contributions

The document lists contributions of 6 major mathematicians - Pythagoras, who discovered the Pythagorean theorem; Aristotle, who studied mathematics in relation to science; Euclid, who wrote the influential Elements on geometry; Al-Khwarizmi, who wrote the first book on algebra; Leonardo da Vinci, who applied geometry to mechanics; and Galileo Galilei, who was a teacher, astrologer, and mathematician.

euclid geometry

Euclid's Geometry is a foundational work in mathematics focused on geometry. It begins with definitions of basic terms like point, line, and plane. Euclid then states postulates and axioms which include that equals added to equals are equal and that the whole is greater than the part. Using these axioms and deductive reasoning, Euclid proves 465 theorems over 13 books, addressing topics like plane geometry, number theory, and solid geometry. The work had a major influence on mathematics for over 2000 years.

3.3 graph systems of linear inequalities

This document discusses how to solve systems of linear inequalities by graphing them. It explains that you first put the inequalities into slope-intercept form, then graph each line as dotted or solid based on the inequality symbols and shade the correct region above or below the line. The solution to the system is the region where the graphs overlap. An example demonstrates these steps to find the solution region between two inequalities.

Proof writing tips

1. The document provides tips for writing proofs, including remembering logical equivalence rules, using definitions, putting new ideas on separate lines for clarity, and using ample space on the page.
2. It emphasizes using different proof styles like direct, contrapositive, and contradiction. An example proof is given using each style.
3. Unpacking definitions is recommended to fully understand the statements and elements involved before beginning a proof. An example problem is worked through to demonstrate this.
4. New ideas should start on new lines to make the logic and flow easier to follow. Using more space on the page also aids readability and allows for feedback.

Linear Inequalities

This document discusses how to graph linear inequalities on a coordinate plane. It explains that less than and greater than signs are represented by dashed lines, while less than or equal to and greater than or equal to signs use solid lines. It notes that inequalities must be written in slope-intercept form before graphing and that the shading depends on whether the inequality involves less than or greater than. Specifically, y greater than inequalities are shaded above the line and y less than inequalities are shaded below the line. Practice graphing many problems to fully understand how to determine the shading direction.

System of linear inequalities

Here is the system of inequalities for the cross-country team problem:
Let x = number of water bottles sold to students
Let y = number of water bottles sold to others
x + y ≤ 100 (they have 100 bottles total)
3x + 5y ≥ 400 (they need at least $400)
Graph the regions defined by these inequalities and the overlapping region is the solution.

Midline theorem - Mathematics - Geometry

The document explains the midline theorem in geometry. The midline theorem states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. The document provides examples of applying the midline theorem to solve for unknown lengths in various triangles. It also includes a short quiz testing understanding of using the midline theorem.

Mathematicians

The document summarizes the contributions of several important early mathematicians, including Pythagoras who discovered the Pythagorean theorem, Aristotle who studied mathematics in relation to science, and Euclid who wrote The Elements which forms the basis of geometry today. It also discusses mathematicians from other cultures like Al-Khwarizmi from Baghdad who wrote the first book on algebra, as well as Western mathematicians such as Galileo, De L'hopital, Euler, and others who made advances in calculus, trigonometry, and mathematical analysis.

5 mathematicans contributions

The document lists contributions of 6 major mathematicians - Pythagoras, who discovered the Pythagorean theorem; Aristotle, who studied mathematics in relation to science; Euclid, who wrote the influential Elements on geometry; Al-Khwarizmi, who wrote the first book on algebra; Leonardo da Vinci, who applied geometry to mechanics; and Galileo Galilei, who was a teacher, astrologer, and mathematician.

euclid geometry

Euclid's Geometry is a foundational work in mathematics focused on geometry. It begins with definitions of basic terms like point, line, and plane. Euclid then states postulates and axioms which include that equals added to equals are equal and that the whole is greater than the part. Using these axioms and deductive reasoning, Euclid proves 465 theorems over 13 books, addressing topics like plane geometry, number theory, and solid geometry. The work had a major influence on mathematics for over 2000 years.

3.3 graph systems of linear inequalities

This document discusses how to solve systems of linear inequalities by graphing them. It explains that you first put the inequalities into slope-intercept form, then graph each line as dotted or solid based on the inequality symbols and shade the correct region above or below the line. The solution to the system is the region where the graphs overlap. An example demonstrates these steps to find the solution region between two inequalities.

Proof writing tips

1. The document provides tips for writing proofs, including remembering logical equivalence rules, using definitions, putting new ideas on separate lines for clarity, and using ample space on the page.
2. It emphasizes using different proof styles like direct, contrapositive, and contradiction. An example proof is given using each style.
3. Unpacking definitions is recommended to fully understand the statements and elements involved before beginning a proof. An example problem is worked through to demonstrate this.
4. New ideas should start on new lines to make the logic and flow easier to follow. Using more space on the page also aids readability and allows for feedback.

Linear Inequalities

This document discusses how to graph linear inequalities on a coordinate plane. It explains that less than and greater than signs are represented by dashed lines, while less than or equal to and greater than or equal to signs use solid lines. It notes that inequalities must be written in slope-intercept form before graphing and that the shading depends on whether the inequality involves less than or greater than. Specifically, y greater than inequalities are shaded above the line and y less than inequalities are shaded below the line. Practice graphing many problems to fully understand how to determine the shading direction.

System of linear inequalities

Here is the system of inequalities for the cross-country team problem:
Let x = number of water bottles sold to students
Let y = number of water bottles sold to others
x + y ≤ 100 (they have 100 bottles total)
3x + 5y ≥ 400 (they need at least $400)
Graph the regions defined by these inequalities and the overlapping region is the solution.

Midline theorem - Mathematics - Geometry

The document explains the midline theorem in geometry. The midline theorem states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. The document provides examples of applying the midline theorem to solve for unknown lengths in various triangles. It also includes a short quiz testing understanding of using the midline theorem.

2.7.1 Congruent Triangles

* Write and interpret congruence statements
* Use properties of congruent triangles
* Prove triangles congruent using the definition of congruence

3.9.1 Dilation, Scale Factor, and Proportion

- Identify dilations and scale factors and use them to solve problems
- Write and simplify ratios and use proportions to solve problems
- Key concepts covered include: dilations, scale factors, ratios, proportions, and using them to solve problems involving transformations and ratios of quantities

Solution of linear equation & inequality

The document discusses solving linear equalities and inequalities with one variable. It defines key terms like equations, inequalities, and linear equations. It then provides steps for solving different types of linear equations and inequalities by collecting like terms, adding/subtracting the variable term to one side, and multiplying/dividing both sides by constants. The document also explains how to graph solutions to inequalities on a number line, indicating open and closed circles based on the inequality symbols. Examples are provided of solving and graphing various linear equalities and inequalities with one variable.

Types of angles

The document defines and provides examples of the five main types of angles: right angles measure 90 degrees, acute angles are less than 90 degrees, obtuse angles are greater than 90 degrees but less than 180 degrees, straight angles measure 180 degrees, and reflex angles are greater than 180 degrees but less than 360 degrees. Examples are given of each type of angle to illustrate how to identify them.

Classifying Angles

This document discusses different types of angles including acute, obtuse, right, and straight angles. It defines an angle as being formed by two rays sharing an endpoint called the vertex. Angles are measured in degrees, with acute angles between 0-90 degrees, obtuse angles between 90-180 degrees, right angles equal to 90 degrees, and a straight angle equaling 180 degrees. It includes examples of each type of angle and encourages identifying them in a game.

Angles powerpoint

The document defines and provides examples of right, acute, and obtuse angles. Right angles measure 90 degrees. Acute angles are less than 90 degrees, while obtuse angles are greater than 90 degrees but less than 180 degrees. Multiple drawings of angles are provided and identified as being either right, acute, or obtuse.

Lines and angles

For those who need help in PPT's for Lines and Angles and want to get good results.
Visit my website :- http://www.soumyamodakbed.blogspot.in/ for more information.

Angles ppt

1) The document defines an angle as being formed by two rays with a common endpoint called the vertex. Angles can have points in their interior, exterior, or on the angle.
2) There are three main ways to name an angle: using three points with the vertex in the middle, using just the vertex point when it is the only angle with that vertex, or using a number within the angle.
3) There are four types of angles: acute, right, obtuse, and straight. Angles are measured in degrees with a full circle being 360 degrees.

Diabetes

There are four main types of diabetes: type 1, type 2, gestational diabetes, and prediabetes. Type 1 is an autoimmune disease where the body attacks the pancreas, type 2 is caused by the body not producing enough insulin or cells not responding to insulin properly. Gestational diabetes occurs during pregnancy. Prediabetes means blood sugar levels are higher than normal but not high enough to be classified as diabetes. Complications of diabetes include cardiovascular, kidney, nerve and eye diseases. Diabetes is tested through A1C, fasting plasma glucose, and oral glucose tolerance tests. Treatment options include oral medications, insulin injections, surgery, and lifestyle changes like diet and exercise. Future treatments may include gene therapy for type 1

Polynomials

This document provides an overview of polynomials, including:
- Defining polynomials as expressions involving variables and coefficients using addition, subtraction, multiplication, and exponents.
- Discussing the history of polynomial notation pioneered by Descartes.
- Explaining the different types of polynomials like monomials, binomials, and trinomials.
- Outlining common uses of polynomials in mathematics, science, and other fields.
- Describing how to find the degree of a polynomial and graph polynomial functions.
- Explaining arithmetic operations like addition, subtraction, and division that can be performed on polynomials.

Advertising

Advertising is a paid, non-personal form of communication used to promote ideas, products, or services. The objectives of advertising include introducing new products to convince customers to try them, retaining existing customers, and winning back customers who have switched to competitors. Advertising provides benefits like employment opportunities and economic growth while allowing consumers to learn about and compare products. However, critics argue that advertising increases product prices and can confuse consumers with similar claims. Overall, while some criticisms exist, advertising plays an important role in modern business by facilitating communication with customers.

old age

This presentation discusses the importance of elders and issues they face. Elders share their life experiences and promote cultural values, acting as mentors. However, they often face neglect from busy children, loneliness, abuse, hopelessness, and helplessness. To help, the government launched pension programs, there are awareness days for elder abuse, and we can increase awareness, education, respite care, counseling, and celebrate grandparents. The presentation concludes by advocating for letting elders age gracefully with dignity and respect.

Chemistry

Osmosis is the movement of solvent molecules through a semi-permeable membrane from an area of higher solvent concentration to lower solvent concentration. The document defines hypertonic, isotonic, and hypotonic solutions and explains how osmosis causes cell shrinkage or swelling depending on the solution. It also provides examples of how osmosis is used in desalination, food concentration, and dairy concentration processes.

Diabetes mellitus

There are four main types of diabetes: type 1, type 2, gestational diabetes, and prediabetes. Type 1 is an autoimmune disease where the body attacks the pancreas, type 2 is caused by the body not producing enough insulin or cells not responding to insulin properly. Gestational diabetes occurs during pregnancy. Prediabetes means blood sugar levels are higher than normal but not high enough to be classified as diabetes. Complications of diabetes include cardiovascular, kidney, nerve and eye diseases. Diabetes is tested through A1C, fasting plasma glucose, and oral glucose tolerance tests. Treatment options include oral medications, insulin injections, surgery, and lifestyle changes like diet and exercise. Future treatments may include gene therapy for type 1

WATER CRISIS “Prediction of 3rd world war”

The document discusses the global water crisis and issues around water management. It notes that water scarcity is increasing due to rising populations and demands for water exceeding supply. The document also discusses historical water management practices, current issues like decreasing groundwater levels, and calls for sustainable water management through conservation efforts, innovative practices, and ensuring access to safe drinking water for all. Scientists warn that without addressing water shortages, wars may be fought over water in the future and ecosystems will suffer serious damage.

Issue of Shares

The document discusses key differences between private and public companies. It states that private companies have restrictions on the number of members and cannot invite the public to subscribe to its shares, while public companies can have an unlimited number of members and can invite public subscription. Additionally, private companies have restrictions on the transfer of shares while public companies do not.

My experience my values 7th

The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.

Solid

The document summarizes key concepts about crystalline and amorphous solids, including:
- Crystalline solids exhibit long-range order while amorphous solids only have short-range order.
- Ionic crystals like NaCl and CsCl form face-centered cubic or body-centered cubic structures to maximize interactions between oppositely charged ions.
- The cohesive energy of ionic crystals can be calculated by considering contributions from Coulomb attraction, electron overlap repulsion, ionization energies, and electron affinities.

Smart class

This document discusses the factors that influence demand, including price, income, and the prices of substitute and complementary goods. It defines quantity demanded as a change in demand due to a change in price, while a change in demand is due to other factors. A demand schedule shows the inverse relationship between price and quantity demanded in a table. A demand curve graphs this relationship, with quantity demanded on the y-axis and price on the x-axis. Movement along the demand curve occurs when quantity demanded changes due to a change in price.

S107

Cancer immunotherapy harnesses the body's immune system to fight cancer by developing antibodies or T cells that target and destroy cancer cells. Researchers are working to develop new immunotherapy approaches that use antibodies or T cells to identify and eliminate cancer cells while leaving normal cells unharmed. Immunotherapy holds promise as a revolutionary new approach for treating many cancer types by giving the body's own immune system tools to fight the disease.

S104

Solid is a state of matter where particles are arranged closely together. Constituent particles in solids can be atoms, molecules, or ions arranged in a definite pattern with no definite shape or volume. There are different types of unit cell structures that describe the arrangement of particles in solids, including primitive, body-centered, and face-centered unit cells. Close packing of spheres in two and three dimensions results in different crystal structures depending on how additional layers are arranged in the voids between lower layers.

2.7.1 Congruent Triangles

* Write and interpret congruence statements
* Use properties of congruent triangles
* Prove triangles congruent using the definition of congruence

3.9.1 Dilation, Scale Factor, and Proportion

- Identify dilations and scale factors and use them to solve problems
- Write and simplify ratios and use proportions to solve problems
- Key concepts covered include: dilations, scale factors, ratios, proportions, and using them to solve problems involving transformations and ratios of quantities

Solution of linear equation & inequality

The document discusses solving linear equalities and inequalities with one variable. It defines key terms like equations, inequalities, and linear equations. It then provides steps for solving different types of linear equations and inequalities by collecting like terms, adding/subtracting the variable term to one side, and multiplying/dividing both sides by constants. The document also explains how to graph solutions to inequalities on a number line, indicating open and closed circles based on the inequality symbols. Examples are provided of solving and graphing various linear equalities and inequalities with one variable.

Types of angles

The document defines and provides examples of the five main types of angles: right angles measure 90 degrees, acute angles are less than 90 degrees, obtuse angles are greater than 90 degrees but less than 180 degrees, straight angles measure 180 degrees, and reflex angles are greater than 180 degrees but less than 360 degrees. Examples are given of each type of angle to illustrate how to identify them.

Classifying Angles

This document discusses different types of angles including acute, obtuse, right, and straight angles. It defines an angle as being formed by two rays sharing an endpoint called the vertex. Angles are measured in degrees, with acute angles between 0-90 degrees, obtuse angles between 90-180 degrees, right angles equal to 90 degrees, and a straight angle equaling 180 degrees. It includes examples of each type of angle and encourages identifying them in a game.

Angles powerpoint

The document defines and provides examples of right, acute, and obtuse angles. Right angles measure 90 degrees. Acute angles are less than 90 degrees, while obtuse angles are greater than 90 degrees but less than 180 degrees. Multiple drawings of angles are provided and identified as being either right, acute, or obtuse.

Lines and angles

For those who need help in PPT's for Lines and Angles and want to get good results.
Visit my website :- http://www.soumyamodakbed.blogspot.in/ for more information.

Angles ppt

1) The document defines an angle as being formed by two rays with a common endpoint called the vertex. Angles can have points in their interior, exterior, or on the angle.
2) There are three main ways to name an angle: using three points with the vertex in the middle, using just the vertex point when it is the only angle with that vertex, or using a number within the angle.
3) There are four types of angles: acute, right, obtuse, and straight. Angles are measured in degrees with a full circle being 360 degrees.

2.7.1 Congruent Triangles

2.7.1 Congruent Triangles

3.9.1 Dilation, Scale Factor, and Proportion

3.9.1 Dilation, Scale Factor, and Proportion

Solution of linear equation & inequality

Solution of linear equation & inequality

Types of angles

Types of angles

Classifying Angles

Classifying Angles

Angles powerpoint

Angles powerpoint

Lines and angles

Lines and angles

Angles ppt

Angles ppt

Diabetes

There are four main types of diabetes: type 1, type 2, gestational diabetes, and prediabetes. Type 1 is an autoimmune disease where the body attacks the pancreas, type 2 is caused by the body not producing enough insulin or cells not responding to insulin properly. Gestational diabetes occurs during pregnancy. Prediabetes means blood sugar levels are higher than normal but not high enough to be classified as diabetes. Complications of diabetes include cardiovascular, kidney, nerve and eye diseases. Diabetes is tested through A1C, fasting plasma glucose, and oral glucose tolerance tests. Treatment options include oral medications, insulin injections, surgery, and lifestyle changes like diet and exercise. Future treatments may include gene therapy for type 1

Polynomials

This document provides an overview of polynomials, including:
- Defining polynomials as expressions involving variables and coefficients using addition, subtraction, multiplication, and exponents.
- Discussing the history of polynomial notation pioneered by Descartes.
- Explaining the different types of polynomials like monomials, binomials, and trinomials.
- Outlining common uses of polynomials in mathematics, science, and other fields.
- Describing how to find the degree of a polynomial and graph polynomial functions.
- Explaining arithmetic operations like addition, subtraction, and division that can be performed on polynomials.

Advertising

Advertising is a paid, non-personal form of communication used to promote ideas, products, or services. The objectives of advertising include introducing new products to convince customers to try them, retaining existing customers, and winning back customers who have switched to competitors. Advertising provides benefits like employment opportunities and economic growth while allowing consumers to learn about and compare products. However, critics argue that advertising increases product prices and can confuse consumers with similar claims. Overall, while some criticisms exist, advertising plays an important role in modern business by facilitating communication with customers.

old age

This presentation discusses the importance of elders and issues they face. Elders share their life experiences and promote cultural values, acting as mentors. However, they often face neglect from busy children, loneliness, abuse, hopelessness, and helplessness. To help, the government launched pension programs, there are awareness days for elder abuse, and we can increase awareness, education, respite care, counseling, and celebrate grandparents. The presentation concludes by advocating for letting elders age gracefully with dignity and respect.

Chemistry

Osmosis is the movement of solvent molecules through a semi-permeable membrane from an area of higher solvent concentration to lower solvent concentration. The document defines hypertonic, isotonic, and hypotonic solutions and explains how osmosis causes cell shrinkage or swelling depending on the solution. It also provides examples of how osmosis is used in desalination, food concentration, and dairy concentration processes.

Diabetes mellitusThere are four main types of diabetes: type 1, type 2, gestational diabetes, and prediabetes. Type 1 is an autoimmune disease where the body attacks the pancreas, type 2 is caused by the body not producing enough insulin or cells not responding to insulin properly. Gestational diabetes occurs during pregnancy. Prediabetes means blood sugar levels are higher than normal but not high enough to be classified as diabetes. Complications of diabetes include cardiovascular, kidney, nerve and eye diseases. Diabetes is tested through A1C, fasting plasma glucose, and oral glucose tolerance tests. Treatment options include oral medications, insulin injections, surgery, and lifestyle changes like diet and exercise. Future treatments may include gene therapy for type 1

WATER CRISIS “Prediction of 3rd world war”

The document discusses the global water crisis and issues around water management. It notes that water scarcity is increasing due to rising populations and demands for water exceeding supply. The document also discusses historical water management practices, current issues like decreasing groundwater levels, and calls for sustainable water management through conservation efforts, innovative practices, and ensuring access to safe drinking water for all. Scientists warn that without addressing water shortages, wars may be fought over water in the future and ecosystems will suffer serious damage.

Issue of Shares

The document discusses key differences between private and public companies. It states that private companies have restrictions on the number of members and cannot invite the public to subscribe to its shares, while public companies can have an unlimited number of members and can invite public subscription. Additionally, private companies have restrictions on the transfer of shares while public companies do not.

My experience my values 7th

The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.

Solid

The document summarizes key concepts about crystalline and amorphous solids, including:
- Crystalline solids exhibit long-range order while amorphous solids only have short-range order.
- Ionic crystals like NaCl and CsCl form face-centered cubic or body-centered cubic structures to maximize interactions between oppositely charged ions.
- The cohesive energy of ionic crystals can be calculated by considering contributions from Coulomb attraction, electron overlap repulsion, ionization energies, and electron affinities.

Smart class

This document discusses the factors that influence demand, including price, income, and the prices of substitute and complementary goods. It defines quantity demanded as a change in demand due to a change in price, while a change in demand is due to other factors. A demand schedule shows the inverse relationship between price and quantity demanded in a table. A demand curve graphs this relationship, with quantity demanded on the y-axis and price on the x-axis. Movement along the demand curve occurs when quantity demanded changes due to a change in price.

S107

Cancer immunotherapy harnesses the body's immune system to fight cancer by developing antibodies or T cells that target and destroy cancer cells. Researchers are working to develop new immunotherapy approaches that use antibodies or T cells to identify and eliminate cancer cells while leaving normal cells unharmed. Immunotherapy holds promise as a revolutionary new approach for treating many cancer types by giving the body's own immune system tools to fight the disease.

S104

Solid is a state of matter where particles are arranged closely together. Constituent particles in solids can be atoms, molecules, or ions arranged in a definite pattern with no definite shape or volume. There are different types of unit cell structures that describe the arrangement of particles in solids, including primitive, body-centered, and face-centered unit cells. Close packing of spheres in two and three dimensions results in different crystal structures depending on how additional layers are arranged in the voids between lower layers.

S103

This document summarizes different types of work, energy, and collisions. It defines positive, negative, and zero work done based on the angle between the applied force and displacement. Kinetic energy is defined as the energy of motion, while potential energy is the energy due to position. The equations for kinetic and potential energy are provided. Conservation of energy and conservative versus non-conservative forces are also summarized. Kinetic energy and total linear momentum are stated to be conserved, while inelastic collisions are defined as those where the initial and final kinetic energies are not equal.

S102

This document discusses the structure and function of neurons and the nervous system. It describes the key parts of a neuron including the cell body, dendrites, and axon. It explains how nerve impulses are transmitted across synapses using neurotransmitters. The document then provides an overview of the major structures of the brain including the cerebrum, cerebellum, brainstem, and how the brain acts as the command center to integrate sensory input and control coordination.

S101

The document provides tips for conserving electricity and reducing energy costs, such as turning off lights and appliances when not in use, using more efficient appliances like CFL bulbs and gas water heaters, avoiding unnecessary opening of refrigerators, not leaving devices on standby, doing laundry in batches, and installing solar panels. It also notes that Energy Star certified devices use 20-30% less energy.

Projectile motion

Projectile motion can be thought of as the combination of horizontal and vertical motion. The horizontal motion is unaffected by gravity and follows the equation x=u*cosθ*t, while the vertical motion is affected by gravity and follows the equation y=u*sinθ*t - 0.5*g*t^2. The path of a projectile forms a parabolic curve. Factors like the launch angle and initial velocity determine the time of flight, maximum height, and horizontal range of the projectile. Projectile motion has applications in various sports to calculate the optimal angle for maximum distance.

Mansi

Anees Jung's book Lost Spring focuses on stories of children from deprived backgrounds who face difficult circumstances like being kidnapped and forced to work in the carpet industry. Some children are maltreated by alcoholic fathers, married off early, or sexually abused. In this chapter, Jung expresses concern over the exploitation of children forced to do hazardous jobs like bangle making and rag picking due to poverty and traditions. This results in the loss of childhood, education, and opportunities. One story focuses on Mukesh, who lives with his brother doing bangle making in Firozabad but wants a different career, though people believe their fate is predetermined by their previous birth. The theme of the book and awards is to create awareness about child

Gravitation

1. The document summarizes Newton's theory of gravitation and key concepts in physics such as the four fundamental forces, Newton's law of universal gravitation, and gravitational potential energy.
2. It also discusses Einstein's theory of general relativity, which improved on Newton's theory by accounting for the fact that gravity propagates at the speed of light rather than instantaneously.
3. The principle of equivalence formed the basis for general relativity and implies that accelerated frames are equivalent to gravitational fields.

Diabetes

Diabetes

Polynomials

Polynomials

Advertising

Advertising

old age

old age

Chemistry

Chemistry

Diabetes mellitus

Diabetes mellitus

WATER CRISIS “Prediction of 3rd world war”

WATER CRISIS “Prediction of 3rd world war”

Issue of Shares

Issue of Shares

My experience my values 7th

My experience my values 7th

Solid

Solid

Smart class

Smart class

S107

S107

S104

S104

Mera anubhav meri siksha 7th

Mera anubhav meri siksha 7th

S103

S103

S102

S102

S101

S101

Projectile motion

Projectile motion

Mansi

Mansi

Gravitation

Gravitation

- 1. TWO SIMPLE PROOFS OF PYTHAGORAS THEOREM Presented by : Dr. Anant W. Vyawahare, Nagpur. 1India's Contribution to Geometry5/8/2015
- 2. • These proofs of Pythagoras theorem are given by Ganesh Daiwadnya. • Ganesh Daiwadnya, an astronomer and a mathematician, was born in 1507 ad. at a place Nandgaon, in Maharashtra, 40 kms . South of Mumbai. 2India's Contribution to Geometry5/8/2015
- 3. The texts written by Ganesh Daiwaidnya ,all in Marathi language, are: • Laghu & Bruhat Tithi Chintamani, • Buddhi-Vilasini – a commentary on Bhaskara`s Lilavati, • A commentary on Bhaskara`s Sidhanta Shiromani 3India's Contribution to Geometry5/8/2015
- 4. • Vrindavana- Tika- a text containing elementary puzzles, • Graha-Laghava- a treatise on astronomy, ( His father, Keshava, an observational astronomer, wrote a book ,Graha Kautaka. was ,then, the only text available in Marathi language) • Shradha Nirnay, Parva Nirnay, etc. 4India's Contribution to Geometry5/8/2015
- 5. • The proofs given below are from Buddhivilasini – a commentary on Bhaskara`s Lilavati ( 1545 ad.) • Of all the proofs available, these proofs seem to be simple. 5India's Contribution to Geometry5/8/2015
- 6. • First proof: • Consider a right angled triangle ABC, right angled at A with base BC. • To prove, (BC)2 = (AB)2 + (AC)2 6India's Contribution to Geometry5/8/2015 C A B
- 7. • Draw a perpendicular AD from A to meet BC in D. • Consider three right angle triangles ABD, ADC, and ABC. 7India's Contribution to Geometry5/8/2015 C A B D
- 8. • Three triangles are similar because • (i) a right angle is common in all the three triangles, • (ii) angle B is same in triangles ABD and ABC, • (iii) angle C is same in triangle ADC and ABC . 8India's Contribution to Geometry5/8/2015
- 9. • Now, triangles ABD Ξ ABC, • Hence, BD /AB = AB / BC, 9India's Contribution to Geometry5/8/2015 C A B D Therefore, BD = (AB)2 / BC, ……………… (1)
- 10. • Similarly, triangles ADC ≡ ABC, 10India's Contribution to Geometry5/8/2015 This gives, DC / AC = AC / BC Hence, DC = (AC)2 / BC ……………………(2) C A B D
- 11. • Add (1) and (2) we get, BD +DC = (AB)2 / BC + (AC)2 / BC • That is, BC = (AB)2 / BC + (AC)2 / BC, Or, (BC)2 = (AB)2 + (AC)2 • This completes the proof. 11India's Contribution to Geometry5/8/2015
- 12. • Another proof: In triangles ABC and ACD, Cos θ = AC/BC = CD/AC, This gives AC2 = BC.CD,…………………………1 Similarly, In triangles ABC and ABD, Sin θ = AB/BC = BD/AB, This gives AB2 = BC.BD,…………………………2 12India's Contribution to Geometry5/8/2015
- 13. • Add (1) and (2) AB2 + AC2 = BC(BD + CD), =BC.BC =BC2, This completes another proof. 13India's Contribution to Geometry5/8/2015
- 14. • Ganesh Daiwaidnya also invented two Pythagorean triplets: (m2 – n2, 2mn, m2 + n2 ), and (p2 – q2, 2pq , p2 + q2 ), and invented a new triplet [(m2 – n2)(p2 – q2) , 2mn (p2 – q2), (m2 + n2)(p2 – q2), ] 14India's Contribution to Geometry5/8/2015
- 15. References: • T.S. Bhanu Murthy: A modern introduction to ancient Indian Mathematics.( Wiley Eastern Publ. , Singapore, 1992) • T.A.Saraswati Amma: Geometry in ancient and medieval India,( Motilal Banarasi Das publ., New Delhi, 1999) • Dr. S. Bhalachandra Rao: Indian Mathematics and Astronomy,(Jnana Deep Publications, Banglore, 1994) 15India's Contribution to Geometry5/8/2015
- 16. THANK YOU 16India's Contribution to Geometry5/8/2015