King John of England signed the Magna Carta in 1215 in response to demands from barons who were unhappy with his harsh rule and frequent tax increases without consultation. The document established certain legal rights and liberties for "freemen" in England. While much of its original text is no longer valid law, clause 29 guaranteeing due process of law is still cited, and the Magna Carta came to symbolize principles of democracy, human rights, and limits on governmental power that have been highly influential worldwide, especially in the development of legal systems like those of the United States and European Convention on Human Rights.
The Magna Carta was signed in 1215 between the barons of England and King John, establishing limits on royal power. It was one of the most important documents of medieval England, influencing later constitutions like the U.S. Constitution. The Magna Carta established principles like rule of law and limitations on rulers' power, helping define modern legal and political systems.
Magna Carta is a 1215 charter agreed to by King John of England that promised protection of church rights, protection from illegal imprisonment for barons, access to swift justice, and limitations on fees paid to the crown. It established principles of due process and equality before the law. Over 800 years, the key clauses concerning liberty and justice in Magna Carta took on greater authority. It is widely viewed as an important early legal document in the development of democracy and establishment of freedom. The 800th anniversary of Magna Carta in 2015 was commemorated through various events around the world.
Washington was founded in 1796 and became the capital of the USA in 1800. It is home to important landmarks like the Capitol building where Congress meets, the White House which has 132 rooms and houses the American president, and the Lincoln Memorial, Washington Monument, and Supreme Court. The Washington Monument is a white marble obelisk that stands 169m high in the city.
Pollution is a major global problem, as evidenced by unhealthy water and polluted air. One cause is acid rain, which is formed from acid gases emitted by factories and vehicle exhaust. To address this, factories need better filters and some may need to close if they do not meet standards. Water pollution is also a serious issue, as many rivers, oceans and seas are used to dump sewage. People often take clean air and water for granted without understanding how important they are to health and nature.
Pollution is a major problem around the world, as unhealthy water and dirty air have become common. Acid rain is caused by factories and cars emitting acid gases into the air, with disastrous effects. Many factories are not properly filtered and some may need to close, while fresh water is increasingly polluted as oceans and seas become dumping grounds for sewage. Appreciating clean air and water will only come when it is too late to help the environment.
Pollution is a major global problem, as evidenced by unhealthy water and dirty air. Acid rain is caused by factories and cars emitting acid gases into the air, with disastrous effects. Many factories are not properly filtered and some may need to close to reduce pollution. Waterways have become dumping grounds for sewage as people have failed to consider the importance of clean water and environment. In the future, the importance of fresh air and clean water will be appreciated once it is too late to undo the damage done to nature.
This document discusses several environmental issues including water scarcity, pollution, energy problems, and the declining water levels of Lake Sevan in Armenia. It notes that global water demand is increasing due to population growth, and that desalination is one solution being used. It also discusses how vehicle emissions and industrial pollution cause acid rain that damages forests, and how countries need to transition to more sustainable energy sources like solar and wind to conserve dwindling oil reserves. The document concludes by explaining that Lake Sevan, the largest body of fresh water in Armenia, has seen its water levels sharply reduced since the 1950s due to excessive water usage for energy and irrigation.
Ecology is the scientific study of the relationship between living organisms and their environment. It includes the study of plant and animal populations, communities, and ecosystems. Some key ecological problems are overuse of pesticides, climate change, urban development, global warming, genetically modified foods, toxic waste, and issues around renewable energy and recycling.
Every day people pollute clean water without caring about their future health or that of animals. Dirty water is dangerous and solving the issue of water pollution seems impossible as some will continue polluting the environment and water sources. Minor solutions include not littering clean water and throwing out garbage. The pollution problem also affects Lake Sevan as fish are becoming ill and the lake is experiencing waterlogging and dirty water according to environmental organizations. While water pollution of Lake Sevan is a global issue, Armenian citizens are responsible for its health and it is important to raise discussions and implement education on environmental protection.
King John of England signed the Magna Carta in 1215 in response to demands from barons who were unhappy with his harsh rule and frequent tax increases without consultation. The document established certain legal rights and liberties for "freemen" in England. While much of its original text is no longer valid law, clause 29 guaranteeing due process of law is still cited, and the Magna Carta came to symbolize principles of democracy, human rights, and limits on governmental power that have been highly influential worldwide, especially in the development of legal systems like those of the United States and European Convention on Human Rights.
The Magna Carta was signed in 1215 between the barons of England and King John, establishing limits on royal power. It was one of the most important documents of medieval England, influencing later constitutions like the U.S. Constitution. The Magna Carta established principles like rule of law and limitations on rulers' power, helping define modern legal and political systems.
Magna Carta is a 1215 charter agreed to by King John of England that promised protection of church rights, protection from illegal imprisonment for barons, access to swift justice, and limitations on fees paid to the crown. It established principles of due process and equality before the law. Over 800 years, the key clauses concerning liberty and justice in Magna Carta took on greater authority. It is widely viewed as an important early legal document in the development of democracy and establishment of freedom. The 800th anniversary of Magna Carta in 2015 was commemorated through various events around the world.
Washington was founded in 1796 and became the capital of the USA in 1800. It is home to important landmarks like the Capitol building where Congress meets, the White House which has 132 rooms and houses the American president, and the Lincoln Memorial, Washington Monument, and Supreme Court. The Washington Monument is a white marble obelisk that stands 169m high in the city.
Pollution is a major global problem, as evidenced by unhealthy water and polluted air. One cause is acid rain, which is formed from acid gases emitted by factories and vehicle exhaust. To address this, factories need better filters and some may need to close if they do not meet standards. Water pollution is also a serious issue, as many rivers, oceans and seas are used to dump sewage. People often take clean air and water for granted without understanding how important they are to health and nature.
Pollution is a major problem around the world, as unhealthy water and dirty air have become common. Acid rain is caused by factories and cars emitting acid gases into the air, with disastrous effects. Many factories are not properly filtered and some may need to close, while fresh water is increasingly polluted as oceans and seas become dumping grounds for sewage. Appreciating clean air and water will only come when it is too late to help the environment.
Pollution is a major global problem, as evidenced by unhealthy water and dirty air. Acid rain is caused by factories and cars emitting acid gases into the air, with disastrous effects. Many factories are not properly filtered and some may need to close to reduce pollution. Waterways have become dumping grounds for sewage as people have failed to consider the importance of clean water and environment. In the future, the importance of fresh air and clean water will be appreciated once it is too late to undo the damage done to nature.
This document discusses several environmental issues including water scarcity, pollution, energy problems, and the declining water levels of Lake Sevan in Armenia. It notes that global water demand is increasing due to population growth, and that desalination is one solution being used. It also discusses how vehicle emissions and industrial pollution cause acid rain that damages forests, and how countries need to transition to more sustainable energy sources like solar and wind to conserve dwindling oil reserves. The document concludes by explaining that Lake Sevan, the largest body of fresh water in Armenia, has seen its water levels sharply reduced since the 1950s due to excessive water usage for energy and irrigation.
Ecology is the scientific study of the relationship between living organisms and their environment. It includes the study of plant and animal populations, communities, and ecosystems. Some key ecological problems are overuse of pesticides, climate change, urban development, global warming, genetically modified foods, toxic waste, and issues around renewable energy and recycling.
Every day people pollute clean water without caring about their future health or that of animals. Dirty water is dangerous and solving the issue of water pollution seems impossible as some will continue polluting the environment and water sources. Minor solutions include not littering clean water and throwing out garbage. The pollution problem also affects Lake Sevan as fish are becoming ill and the lake is experiencing waterlogging and dirty water according to environmental organizations. While water pollution of Lake Sevan is a global issue, Armenian citizens are responsible for its health and it is important to raise discussions and implement education on environmental protection.
2. Ուղղանկյուն
եռանկյան ներքնաձիգի
քառակուսին հավասար է էջերի
քառակուսիների գումարին:
3. Պյութագորասի
հակադարձ թեորեմ
Եթե եռանկյան մի կողմի քառակուսին
հավասար է մյուս երկու կողմերի քառակուսիների
գումարին, ապա այդ
եռանկյունը ուղղանկյուն եռանկյուն է:
A
c
a
C B
b
4. Ապացույց 1.
Եռանկյանը լրացնենք մինչև a+b կողմերով
քառակուսի այնպես,ինչպես ցույց է տրված
նկարում:Այս քառակուսու մակերեսը հավասար
է 2
(a+b) :Մյուս կողմից,այդ քառակուսին
կազմված է չորս հավասար
եռանկյուններից,որոնցից յուրաքանչյուրի
մակերեսը հավասար է 1:2*ab,և c կողմով
քառակուսուց,ուստի
2 2
S=4*(1:2)*ab+c=2ab+c
Այսպիսով c քառակուսին հավասար է aքառակուսի + b
c քառակուսի:
b Թեորեմը ապացուցված է
a
5. Ապացույց 2.
Դիցուք CD-ն ABC եռանկյան բարձրությունն է
2
AC = AB*AD ,հանգունորեն BC2=BD*AB:Անդամ առ անդամ
գումարելով այս հավասարությունները և հաշվի առնելով,որ
AD+DB=AB,ստանում ենք
C
2 2 2
AC+BC=AD*AB+BD*AB=(AD+DB)AB=AB
C
A D B
6. Ապացույց 3.
Դիցուք ABC-ն տրված ուղղանկյուն եռանկյունն է C ուղիղ անկյունով:C
ուղիղ անկյան գագաթից տանենք CD բարձրություն:
Ըստ անկյան կոսինուսի սահմանման ` cosA=AD =AC
AC AB
cosB=BD=BC
BC AB
2 2 2
Գումարելով ստացված հավասարությունները և հաշվի առնելով,որ
AD+DB=AB կստանանք
AC+BC=AB(AD+DB)=AB
7. Ա
Ապացուցում 3.
Թեորեմը կարելի է ապացուցել՝
օգտվելով վեկտորների գումարման կանոնից և
վեկտորների սկալյար արտադրյալի բանաձևից:
Նկարից երևում է, որ
Հաշվի
առնելով դա՝
Այսպիսով, ստացանք
որը համարժեք է կոսինուսների թեորեմի
հավասարմանը։
8. Ապացույց 4.
c կողմին
ուղղահայ տարեք
նկարում․ այդ
դեպքում
Երկու կողմերը բազմապատկելով c-
ով՝, կստանաք
Մյուս ուղղահայցները տանելով՝
կստանաք
Վերջին երկու հավասարությունները
գումարելով՝ կստանաք
Առաջին հավասարումը
եկրկորդ հավասարումից
հանելով՝ կստանանք
Որը կարելի է
պարզեցնել
հետևյալ
տեսքի․
9. Ապացույց 5.
Եռանկյան ցանկացած կողմի քառակուսին հավասար
է մյուս երկու կողմերի քառակուսիների գումարին՝
հանած այդ կողմերի և նրանցով կազմված անկյան
կոսինուսի կրկնապատիկ արտադրյալը՝
Այն հանդիսանում է Պյութագորասի
թեորեմի ընդհանրացված տարբերակը։
Երբ γ անկյունը ուղիղ է (90° կամ π/2
ռադիան), կոսինուսների թեորեմը վերածվում է
Պյութագորասի թեորեմին․
Եռանկյան տարբեր կողմերի միջև ընկած
անկյունները ընտրելիս՝ այն կստանա
հետևյալ տեսքը
