Category: CBSE Class 9

On this page, you will find NCERT Class 9 History Chapter 4 Notes Pdf free download. CBSE Class 9 Social Science Notes History Chapter 4 SST Forest Society and Colonialism will seemingly, help them to revise the important concepts in less time.

Forest Society and Colonialism Class 9 Notes Social Science History Chapter 4

CBSE Class 9 History Chapter 4 Notes Understanding the Lesson

1. Forests treasure valuable resources. They provide us many things, such as paper, gum, honey, rubber, fruits, flowers and many other things.

2. Forest area is fast disappearing due to various reasons. The process of deforestation began many centuries ago; but under colonial rule it became more systematic and extensive.

3. In the colonial period, cultivation expanded rapidly for a variety of reasons. The British thought that forests were unproductive. They were considered to be wilderness that had to be brought under cultivation so that the land could yield agricultural products and revenue. This led to the clearance of forests.

4. By the 1830s, trees were felled on a massive scale and vast quantities of timber were exported from India to England.

5. The spread of railways from the 1850s also led to deforestation. Large areas of natural forests were also cleared to make way for tea, coffee and rubber plantations to meet Europe’s growing need for these commodities.

6. In order to manage the forests the British made Dietrich Brandis, a German expert, the first Inspector General of Forests in India. Brandis thought that felling of trees and grazing should be restricted in order to preserve forests for timber production. Hence, rules about the use of forest resources were made. Anybody who cut trees without permission was punished.

7. Brandis introduced ‘scientific forestry’ in which natural forests which had lots of different types of trees were cut down. In their place, one type of tree was planted in straight rows.

8. In 1865, the Forest Act was enacted. It was amended twice, once in 1878 and then in 1927. The 1878 Act divided forests into three categories-reserved, protected and village forests. The best forests were called ‘reserved forests’. Villagers could not take anything from these forests.

9. The Forest Act meant severe hardship for villagers across the country. After the Act, all their everyday practices such as grazing cattle, collecting fruits and roots, etc., became illegal.

10. The British government also banned shifting cultivation. As a result, many communities were forcibly displaced from their homes in the forests. The new forest laws also deprived the forest dwellers of their customary rights to hunt. Those who were caught hunting were now punished for poaching.

11. Some people, however, benefited from the new opportunities that had opened up in trade. Many communities started trading in forest products such as hides, horns silk cocoons, bamboo, spices, fibres, grasses, gums and resins, etc.

12. New opportunities of work did not always mean improved well-being for the people. In Assam, both men and women from forest communities like Santhals and Oraons from Jharkhand, and Gonds from Chhattisgarh were recruited to work on tea plantations. Their wages were low and conditions of work were very bad.

13. In many parts of India, and across the world, forest communities rebelled against the changes that were being imposed on them. The first rebellion took place in Bastar district in Madhya Pradesh in 1910 which was suppressed by the British troops.

14. The colonial power in Indonesia were the Dutch and there were many similarities in the laws for forest control in Indonesia and India. Java in Indonesia is where the Dutch started forest management. Like the British, they wanted timber from Java to build ships.

15. The Dutch enacted forest laws in Java, restricting villagers access to forests. They were punished for grazing cattle in young stands, transporting wood without a permit, or travelling on forest roads with horse carts or cattle.

16. Around 1890, Surontiko Samin of Randublating village, a teak forest village, began questioning state ownership of the forest. He argued that the state could not own wind, water, earth and wood. Soon a widespread movement developed.

17. War and deforestation are closely related. The two world wars had a major impact on forests. In India, the forest department cut trees freely to meet British war needs. In Java, just before the Japanese occupied the region, the Dutch followed ‘a scorched earth’ policy, destroying sawmills, and burning huge piles of giant teak logs so that they would not fall into Japanese hands.

18. Conservation of forests has now become and important goal. The government has recognized that in order to meet this goal, the people who live near the forests must be involved.

Forest Society and Colonialism Class 9 CBSE Notes Important Terms

Deforestation: The disappearance of forests.

Sleepers: Wooden planks laid across railway tracks; they hold the tracks in position.

Plantation: Planting of one type of trees in straight rows.

Scientific forestry: A system of cutting trees controlled by the forest department in which old trees are cut and new ones planted.

Poaching: The illegal practice of trespassing on another’s property to hunt or steal game without the landowner’s permission.

Pargana: Cluster of villages.

Notes of History Class 9 Chapter 4 Time Period

1850s: the spread of railways started from this period.

1860s: The railway network expanded rapidly from this period.

1864: The Indian Forest Service was set up.

1865: The Forest Act was enacted.

1878: The Forest Act was amended.

1910: Rebellion took place in the Kingdom of Bastar.

1927: The Forest Act was amended.

On this page, you will find NCERT Class 9 History Chapter 3 Notes Pdf free download. CBSE Class 9 Social Science Notes History Chapter 3 SST Nazism and the Rise of Hitler will seemingly, help them to revise the important concepts in less time.

Nazism and the Rise of Hitler Class 9 Notes Social Science History Chapter 3

CBSE Class 9 History Chapter 3 Notes Understanding the Lesson

1. Germany was a powerful empire in the early twentieth century. It fought the First World War (1914-1918) alongside the Austrian empire and against the Allies.

2. Germany was defeated in 1918 and the emperor was abdicated. This gave an opportunity to parliamentary parties to recast German polity. A National Assembly met at Weimar and established a democratic constitution with a federal structure.

3. German people did not welcome the new Weimar Republic because they held it responsible for the defeat of Germany in the war and the disgrace at Versailles.

4. Germany faced deep economic crisis in 1923. It had fought the war largely on loans and had to pay war reparations in gold. This depleted gold reserves at a time resources were scarce.

5. The German economy was further hit by the Great Economic Depression. By 1932, industrial production was reduced to 40 percent of the 1929 level. Workers lost their jobs. The economic crisis created deep anxiety and fears in people. Since the Weimar Republic failed to manage this crisis; people lost confidence in the democratic parliamentary system.

6. All this formed the background of Hitler’s rise to power. The German defeat horrified Hitler and the Versailles Treaty made him furious. In 1919, he joined the German Worker’s Party. He subsequently took over the organization and renamed it the National Socialist German Worker’s Party which later came to be known as the Nazi Party.

7. Nazism became a mass movement during the Great Depression. In a situation when workers lost jobs and the middle classes were threatened with destitution, Nazi propaganda stirred hopes in them. In 1928, the Nazi Party was not very successful but by 1932, it had become the largest party. Hitler impressed the German people with his powerful speeches. He promised them to build a strong nation, undo the injustice of the Versailles Treaty and restore their dignity.

8. On 30 January 1933, Hitler was offered the Chancellorship of Germany. Having acquired power, Hilter set out to dismantle the structures of democratic rule. Civic rights like freedom of speech, press and assembly were suspended. The Communists were sent to the newly established concentration camps.

9. The famous Enabling Act, passed on 3 March 1933, established dictatorship in Germany. Hilter became the most powerful man in the country. He sidelined Parliament and rule by decree. He also banned all political parties and trade unions.

10. Special surveillance and security forces were created to control and order society in ways that the Nazis wanted. People could now be detained in Gestapo (secrete state police) torture chambers, rounded up and sent to concentration camps.

11. Hitler got quick success in his foreign policy. He integrated Austria and Germany in 1938 under the slogan, One people, One empire and One leader.

12. In September 1939, Germany invaded Poland. This started a war with France and England. In September 1940, a Tripartite Pact was signed between Germany, Italy and Japan which strengthened Hitler’s claim to international power.

13. Hitler created an exclusive racial community of pure German by eliminating all those who were seen as ‘undesirable’ in extended empire. Nazis wanted only a society of pure and healthy Nordic Aryan’s. Only they were seen as worthy of prospering and multiplying against all who were classed; as ‘undesirables’.

14. Jews, Gypsies and blacks living in Nazi Germany were widely persecuted. Even Russians and Poles were considered subhuman. However, Jews remained the worst sufferers. They were terrorized, pauperized and segregated.

15. Hitler was fanatically interested in the youth of the country. He felt that a strong Nazi society could be established only by teaching children Nazi ideology. So, all schools were ‘cleansed’ and ‘purified’. Jew teachers and children were thrown out of schools.

16. School textbooks were rewritten, racial science was introduced to justify Nazi ideas of race. Children were taught to be loyal and submissive, hate Jews, and worship Hitler.

17. Youth organizations were made responsible for educating German youth in ‘the spirit of National Socialism’. Ten-year-olds had to enter Jungvolk and at 14 all boys had to join the Nazi youth organization e. Hilter youth where they learnt to worship war, glorify aggression and violence, and condemn democracy.

18. In Nazi Germany, all mothers were not treated equally. Women who bore racially undesirable children were punished and those who produced racially desirable children were awarded.

19. The art of propaganda that Nazis used was peculiar. They never used the words ‘kill’ or ‘murder’ in their official communications. Mass killings were termed special treatment, final solution (for the Jews), euthanasia (for the disabled), selection and ‘Evacuation’ meant deporting people to gas chambers.

Nazism and the Rise of Hitler Class 9 CBSE Notes Important Terms

Allies: The Allied Powers were initially led by the UK and France. In 1941 they were joined by the USSR and USA. They fought against the Axis Powers, namely Germany, Italy and Japan.

Genocidal: Killing on large scale leading to destruction of large sections of people.

Deplete: Reduce, empty out.

Reparation: Make up for a wrong done.

Wall Street Exchange: The name of the world’s biggest stock exchange located in the USA.

Hyperinflation: A situation when prices rise phenomenally high.

Proletarianization: To become impoverished to the level of working classes.

Propaganda: Specific type of message directly aimed at influencing the opinion of people through the use of posters, films, speeches, etc.

Concentration camp: A camp where people were isolated and detained without due process of law. Typically, it was surrounded by electrified barbed wire fences.

Nordic German Aryans: One branch of those classified as Aryans. They lived in north European countries and had German or related origin.

Gypsy: The groups that were classified as ‘gypsy’ had their own community identity. Sinti and Roma were too such communities. Many of them traced their origin to India.

Pauperised: Reduce to absolute poverty.

Persecution: Systematic, organized punishment of those belonging to a group or religion.

Usurers: Moneylenders charging excessive interest, often used as a term of abuse.

Synagogues: Place of worship for people of Jewish faith.

Jungvolk: Nazi youth groups for children below 14 years of age.

Holocaust: It was a genocide in which Hitler’s Nazi Germany and its collaborators killed about six million Jews.

Notes of History Class 9 Chapter 3 Time Period

1914: The First World War broke out.

1917: Entry of the USA in the First World War.

1918: The First World War ended.

1919: Treaty of Versailles.

1923: Economic crisis in Germany.

1929: The Great Economic Depression started.

1932: The Nazi Party became the largest party in the German Parliament.

1933: Hilter was made Chancellor of Germany.

1939: Germany invaded Poland, Second World War began.

1940: Hitler was at the height of his power.

1941: The USA joined Second World War.

1945: Allied victory in Europe, Hitler was defeated.

On this page, you will find NCERT Class 9 History Chapter 2 Notes Pdf free download. CBSE Class 9 Social Science Notes History Chapter 2 SST Socialism in Europe and the Russian Revolution will seemingly, help them to revise the important concepts in less time.

Socialism in Europe and the Russian Revolution Class 9 Notes Social Science History Chapter 2

CBSE Class 9 History Chapter 2 Notes Understanding the Lesson

1. The powerful ideas of freedom and equality circulated in Europe after the French Revolution. In many parts of the world including Europe and Asia, new ideas about individual rights and who controlled social power began to be discussed.

2. Not everyone in Europe wanted a complete transformation of society. Responses varied from those who accepted that some change was necessary but wished for a gradual shift, to those who wanted to reconstruct society radically. Some were conservatives, others were liberals or radicals. Conservatives were opposed to radicals and liberals. After the French Revolution, however, even conservatives had opened their minds to the need for change.

3. Liberals and radicals opposed to the privileges the old aristocracy had by birth because they firmly believed in aristocracy had by birth because they firmly believed in the value of individual effort, labour and enterprise. So, many working men and women rallied around liberal and radical groups and parties in the early nineteenth century.

4. By the mid-nineteenth century in Europe, socialism attracted widespread attention. Socialist were against private property, and saw it as the root of all social evils of the time. They wanted that attention should be part to collective social interest.

5. Karl Marx was against capitalism. He believed that to free themselves from capitalist exploitation, workers had to construct a radically socialist society where all property was socially controlled. This would be a communist society.

6. Socialists took over the government in Russia in 1917 by a eliminating monarchy there. But this came about after a long struggle.

7. In 1914, Tsar Nicholas II ruled Russia and its empire. At that time, the vast majority of Russia’s people were agriculturists. Industries were few and were controlled by industrialists.

8. Russia was an autocracy. Unlike other European rulers, even at the beginning of the twentieth century, the Tsar was not subject to parliament. Liberals in Russia campaigned to end this system. The year 1904 was bad for Russian workers. Prices of essential goods rose so quickly that real wages declined by 20 percent. Hence, the workers in St. Petersburg went on strike demanding an increase in wages.

9. When the procession of workers reached the Winter Palace it was attacked by the police and the Cossacks. Hundreds of workers were killed and many were wounded. The incident, known as Bloody Sunday, started a series of events that came to be known as the 1905 Revolution.

10. During the 1905 Revolution, the Tsar allowed the creation of an elected consultative Parliament or Duma. But he dismissed the first Duma within 75 days and re-elected second Duma within three months. He did not want any reduction in his power. He changed the voting laws and packed the third Duma with conservative politicians.

11. In 1914, the first world war was broke out. In Russia, the war was initially popular and people rallied around the Tsar. But soon the number of his supporters went down, because he did not pay any heed to the soldier’s welfare.

12. The Russians Revolution of 1917 revolves around two primary events—the February Revolution and the October Revolution. The February Revolution brought down the monarchy in Russia through violent demonstrations and riots on the street of Petrograd.

13. But the February Revolution failed to express the wishes of the majority of the Russian population, as the event was primarily limited to the city of Petrograd.

14. However, most of those who took power after the February Revolution, in the Provisional Government that replaced the Tsar, and in the Petrograd Soviet generally favoured rule that was at least partially democratic.

15. The October Revolution, also called the Bolshevik Revolution, overturned the interim Provisional Government and established the Soviet Union.

16. After October, the Bolsheviks realised that they could not maintain power in an election-based system without sharing power with other parties and compromising their principles. As a result, they formally abandoned the democratic process in January 1918 and declared themselves the representatives of a dictatorship of the proletariat. In response, the Russian civil war broke out which lasted till 1920.

17. During the civil war, the Bolsheviks kept industries and banks nationalized. They permitted the peasants to cultivate the land that had been socialized. They also introduced centralized planning which led to economic growth.

18. The Bolsheviks encouraged colonial peoples to follow their experiment. Many non-Russians participated in the conference of the peoples of the East and the Bolshevik founded Comintern. By the time of the outbreak of the Second World War, the USSR had given socialism a global face but afterwards it began losing its popularity.

Socialism in Europe and the Russian Revolution Class 9 CBSE Notes Important Terms

Conservative: A person who favours free enterprise, private ownership, and holds traditional values.

Radical: A person who advocates thorough or complete political or social reform.

Liberal: A person who is open to new ideas.

Suffragette movement: A movement to give women the right to vote.

Autocracy: A system of government by one person with absolute power.

Jadidists: Muslim reformers within the Russian Empire.

Read wage: Reflects the quantities of goods which the wages will actually buy.

Deported: Forcibly removed from one’s own country.

Exiled: Forced to live away from one’s own country.

Collectivization: Consolidation of individual landholdings and labor into collective farms.

Socialism: A political and economic theory of social organization which advocates that the means of production distribution and exchange should be owned or regulated by the community as a whole.

Kulaks: The name for well-to-do peasants in Russia.

Notes of History Class 9 Chapter 2 Time Period

1850s-1880s: Debates over socialism in Russia.

1898: Formation of the Russian Social Democratic Workers Party.

1905: The Bloody Sunday and the Revolution of 1905.

1917: 2nd March- Abdication of the Tsar.

24th October- Bolshevik uprising in Petrograd.

1918-20: The Civil war in Russia.

1919: Formation of Comintern.

1929: Beginning of Collectivisation.

On this page, you will find NCERT Class 9 History Chapter 1 Notes Pdf free download. CBSE Class 9 Social Science Notes History Chapter 1 SST The French Revolution will seemingly, help them to revise the important concepts in less time.

The French Revolution Class 9 Notes Social Science History Chapter 1

CBSE Class 9 History Chapter 1 Notes Understanding the Lesson

1. In 1774, Louis XVI of the Bourbon family of kings ascended the throne of France. Upon his accession the new king found an empty treasury. So the state was forced to increase taxes.

2. French society in the eighteenth century was divided into three estates, but only the members of the third estate had to pay taxes. The third estate was comprised of big businessmen, merchants, court officials, peasants and artisans. The members of the first two estates, that is, the clergy and the nobility, enjoyed certain privileges by birth.

3. The phenomenal increase in the population of France in 1789 led to a rapid increase in the demand for food grains. As a result the price of bread rose rapidly. It became difficult for the poor to keep pace with the rise in prices. The gap between the poor and the rich widened.

4. In the eighteenth century there emerged new social groups, termed the middle class. The people belonging to this class were educated and believed that no group in society should be privileged by birth. Rather, a person’s social position must depend on this merit.

5. Philosophers such as John Locke, Jean Jacques Rousseau and Montesquieu put forward the ideas of freedom and equal laws and opportunities for all. These philosophers refuted the doctrine of the divine and absolute right of the monarch.

6. At a time when general public was facing hard times, the news that Louis XVI planned to impose further taxes generated anger among them. They protested against the system of privileges. But as Louis XVI had to increase taxes, he called together an assembly of the Estates-General on 5 May 1789, to pass proposals for new taxes.

7. Voting in the Estates-General in the past had been conducted according to the principle that each estate had one vote. This time too Louis XVI was determined to continue the same practice. But the members of the third estate demanded that voting now be conducted by the assembly as a whole, where each member would have one vote. However, the king rejected their proposal which compelled them to walkout.

8. Meanwhile, the representatives of the third estate assembled in the grounds of Versailles on 20 June and declared themselves a National Assembly. They had drafted a constitution for France that would limit the powers of the monarch.

9. In the meantime the condition of the common mass in France worsened because of the bad harvest. Unaware of his subjects problem the king ordered troops to move into Paris on 4 July, the agitated crowed stormed and destroyed Bastille.

10. Louis XVI, could not suppress his revolting subjects and finally accorded recognition to the National Assembly and accepted the principle that his powers would from now on be checked by a constitution.

11. On the night of 4 August 1789, the Assembly passed a decree abolishing the feudal system of obligations and taxes. Members of the clergy too were forced to give up their privileges.

12. The National Assembly completed the draft of the constitution in 1791. Powers were now separated and assigned to different institutions-the legislature, executive and judiciary.

13. The constitution began with a Declaration of the Rights of Man and Citizen. Rights such as the right to life, freedom of speech, freedom of opinion, equality before law, were established as natural rights.

14. But one wrong step of Louis XVI worsened the situation in France. Although he had signed the constitution, he entered into secret negotiations with the king of Prussia. In the meantime, the National Assembly declared war against Prussia and Austria. Thousands of volunteers joined the army. They saw this as a war of the people against kings and aristocracies all over Europe.

15. Large sections of population were in favour of the continuation of revolution, as the constitution of 1791 gave political rights only to the richer sections of society. Political clubs came into existence. They wanted to plan their own forms of action. The most successful of these clubs was that of the Jacobins.

16. The members of the Jacobin club belonged mainly to the less prosperous sections of society. Their leader was Maximilian Robespierre.

17. In the summer of 1792 the Jacobins planned an insurrection of a large number of Parisians who were angered by the short supplies and high prices of food. On the morning of August 10 they stormed the Palace of the Tuileries, killed the king’s guards and held the king himself as hostage. Later, the Assembly imprisoned the royal family.

18. Louis XVI and his queen were sentenced to death in 1793. Thus, monarchy was abolished and France was made a republic. Then started the Reign of Terror in France under the leadership of Robespierre. But it could last for long. In July 1794, Robespierre was arrested and then sentenced to death.

19. The fall of the Jacobin government allowed the wealthier middle classes to seize power. But political instability continued which paved the way for the rise of Napoleon Bonaparte who crowned himself Emperor of France in 1804. He was finally defeated at Waterloo in 1815.

20. The Ideas of liberty and democratic rights were the most important legacy of the French Revolution for the peoples of the world during the nineteenth and twentieth centuries.

The French Revolution Class 9 CBSE Notes Important Terms

Livre: Unit of currency in the France, discontinued in 1794.

Clergy: Group of persons invested with special functions in the church.

Tithe: A tax levied by the church, comprising on tenth of the agricultural produce.

Taille: Tax to be paid directly to the state.

Subsistence: An extreme situation where the basic means of livelihood are endangered.

Chateau: (PI. chateaux)-Castle or stately residence belonging to a king or a nobleman.

Manor: An estate consisting of the lord’s lands and his mansion.

Treason: Betrayal of one’s country or government.

Guillotine: A device consisting of two poles and a blade with which a person is beheaded.

Negroes: A term used for the indigenous people of Africa south of the Sahara.

Emancipation: The act of freeing.

Marseillaise: The national anthem of France.

Sans-culottes: Those without knee-breeches.

Notes of History Class 9 Chapter 1 Time Period

1774: Louis XVI became the king of France.

1789: The Bastille was stormed.

1791: A constitution was framed to limit the power of the king.

1792-93: France became a republic, the king was beheaded.

1804: Napoleon became the emperor of France.

1815: Napoleon was defeated at Waterloo.

On this page, you will find Probability Class 9 Notes Maths Chapter 15 Pdf free download. CBSE NCERT Class 9 Maths Notes Chapter 15 Probability will seemingly help them to revise the important concepts in less time.

CBSE Class 9 Maths Chapter 15 Notes Probability

Probability Class 9 Notes Understanding the Lesson

1. Experiment: A procedure which produces some well-defined possible outcomes.

2. Random experiment: An experiment which when performed produces one of the several possible outcomes called a random experiment.

3. Trial: When we perform an experiment it is called a trial of the experiment.

4. Event: The set of outcomes of an experiment to which probability is assigned.

5. The empirical (or experimental) probability P(E) of an event E is given by
\(\mathrm{P}(\mathrm{E})=\frac{\text { Number of trials in which } \mathrm{E} \text { has happened }}{\text { Total number of trials }}\)
So, Probability of not happening an event \(\mathrm{P}(\overline{\mathrm{E}})\)= 1 – P(E)

6. The probability of an event lies between 0 and 1 (0 and 1 are included).

7. Impossible event: An event which never happens.

8. Certain event : An event which definitely happens.

On this page, you will find Statistics Class 9 Notes Maths Chapter 14 Pdf free download. CBSE NCERT Class 9 Maths Notes Chapter 14 Statistics will seemingly help them to revise the important concepts in less time.

CBSE Class 9 Maths Chapter 14 Notes Statistics

Statistics Class 9 Notes Understanding the Lesson

There are two types of data:

• Primary
• Secondary.

We can represent the data by:

• Ungrouped and
• Grouped frequency distribution.

Data can also be represented by:

• Bar graph
• Histogram
• Frequency polygons.

Class mark of grouped data
\(=\frac{\text { lower limit }+\text { upper limit }}{2}\)

Measure of central tendencies are mean, median and mode.

Mean:

where, Σf_ix_i = Sum of all observations
Σf_i = Total frequency.

Median: Arrange the observations in ascending or descending order,

(i) If numbers of observations (x) are odd, then median \(\left(\frac{n+1}{2}\right)^{t h}\) terms

(ii) If number of observations (x) are even, then median \(\frac{n^{t h}}{2} \text { and }\left(\frac{n}{2}+1\right)^{t h}\)

Mode: The observation whose frequency is highest.

Relationship between mean, median and mode:
Mode = 3 Median – 2 Mean.

Graphical Representation of Data

• Bar graphs: A bar graph is a pictorial representation of the numerical data by a number of bars (rectangles) of uniform width erected horizontally or vertically with equal space between them. Each rectangle or bar represents only one value of the numerical data and the height or length of bar indicates the corresponding value of the numerical data.
• Histogram: A histogram or frequency histogram is a representation of a frequency distribution in the form of rectangles such that there is no gap between any two successive rectangles.
• Frequency polygon: It is another method of representing frequency distribution graphically.

On this page, you will find Constructions Class 9 Notes Maths Chapter 12 Pdf free download. CBSE NCERT Class 9 Maths Notes Chapter 12 Constructions will seemingly help them to revise the important concepts in less time.

CBSE Class 9 Maths Chapter 12 Notes Constructions

Constructions Class 9 Notes Understanding the Lesson

• Geometrical construction means using only a ruler and a pair of compasses as geometrical instruments.
• Protractor may be used for drawing non-standard angles.

Construction of bisector of a line segment using compass

• Draw a line segment, say AB.
• With both the end points (A and B) of line segment as centre and taking radius of more than half of the measure of line segment, draw the arcs on both sides, which cut at two points on opposite side, say P and Q.
  
• Join these two points.
  This line (PQ) is the required bisector.

Construction of the bisector of a given angle (say ∠ABC)

• With A as centre and a small radius draw an arc, cutting AB at P and AC at Q.
• With P as centre and the same radius as above, draw an arc.
• With Q as centre and with same radius, draw another arc, cutting the previous arc at D.
  
• Join AD.

AD is the required bisector of ∠BCA.

• Construction of some Angies and Triangles ;
• Constructions of some standard angles such as 30°, 45°, 60°, 75°, 90°, 120° etc. are possible using a
  ruler and a pair of compasses. :
• Construction of a triangle is possible, when its base, a base angle and the sum of other two sides are given.
• Construction of a triangle is possible, when its base, a base angle and the difference of other two sides are given.
• Construction of a triangle is possible when its perimeter

On this page, you will find Circles Class 9 Notes Maths Chapter 11 Pdf free download. CBSE NCERT Class 9 Maths Notes Chapter 11 Circles will seemingly help them to revise the important concepts in less time.

CBSE Class 9 Maths Chapter 11 Notes Circles

Circles Class 9 Notes Understanding the Lesson

Circle is the collection of all points in a plane, which are equidistant from a fixed point in the plane. The fixed point is called the centre O and the given distance is called the radius r of the circle.

Concentric circles: Circles having same centre and different radii are called concentric circles.

Arc: A continuous piece of a circle is called an arc of the circle.

Chord: A line segment joining any two points on a circle is called the chord of the circle.

Diameter: A chord passing through the centre of a circle is called the diameter of the circle.

• Semicircle: A diameter of a circle divides it into two equal parts which are arc. Each of these two arcs is called semicircle.
• Angle of semicircle is right angle.
• If two arcs are equal, then their corresponding, chords are also equal.

Theorem 10.1: Equal chords of a circle subtend equal angle at the centre of the circle.
Theorem 10.2: If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
Theorem 10.3: The perpendicular drawn from centre to the chord of circle bisects the chord.

Theorem 10.4: The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. Theorem 10.5: There is one and only one circle passing through three non-collinear points.
Theorem 10.6: Equal chords of circle are equidistant from centre.

Theorem 10.7: Chords equidistant from the centre of a circle are equal in length.

• If two circles intersect in two points, then the line through the centres is perpendicular to the common chord.

Theorem 10.8: The angle subtended by an arc at the centre of circle is twice the angle subtended at remaining part of circumference.
Theorem 10.9: Any two angles in the same segment of the circle are equal.
Theorem 10.10: If a line segment joining two points subtends equal angles at two other points on the same side of the line containing the line segment, the four points lie on a circle (i.e., they are concyclic).

Cyclic Quadrilateral: If all the vertices of a quadrilateral lie on the circumference of circle, then quadrilateral is called cyclic.

Theorem 10.11: In a cyclic quadrilateral the sum of opposite angles is 180°.
Theorem 10.12: In a quadrilateral if the sum of opposite angles is 180°, then quadrilateral is cyclic.

• The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

On this page, you will find Areas of Parallelograms and Triangles Class 9 Notes Maths Chapter 10 Pdf free download. CBSE NCERT Class 9 Maths Notes Chapter 10 Areas of Parallelograms and Triangles will seemingly help them to revise the important concepts in less time.

CBSE Class 9 Maths Chapter 9 Notes Areas of Parallelograms and Triangles

Areas of Parallelograms and Triangles Class 9 Notes Understanding the Lesson

1. Area of a parallelogram = base x height
= DC x AE

2. Area of a triangle = \(\frac{1}{2}\)base x height
\(\frac{1}{2}\) BC x AD

3. Area of a trapezium = x (Sum of parallel sides) x Distance between them
\(\frac{1}{2}\) (AB + DC) x AE

4. Area of a rhombus =\(\frac{1}{2}\) \(\frac{1}{2}\) x product of diagonals A B
\(\frac{1}{2}\) x AC x BD

5. Two figures are said to be on the same base and between the same parallels, if they have a common side (base) and the vertices (or the vertex) opposite to the common base of each figure He on a line parallel to the base.

Theorem 9.1: Parallelograms on the same base and between the same parallels are equal in area.
ar(ABCD) = ar(EFCD)

Theorem 9.2: Triangles on the same base and between the same parallels are equal in area.
ar(ΔABC) = ar(ΔPBC)

Theorem 9.3: Two triangles having the same base and equal areas lie between the same parallels.
If a triangle and a parallelogram are on the same base and between the same parallels, then

(i) Area of triangle = \(\frac{1}{2}\) x area of the parallelogram
ar(ΔPDC) = \(\frac{1}{2}\) ar(||gmABCD)

(ii) A diagonal of parallelogram divides it into two triangles of equal areas.
ar(ΔABD) = ar(ΔBCD)

(iii) If each diagonal of a quadrilateral separates it into two triangles of equal area, then the quadrilateral is a parallelogram.

(iv) A median AD of a ΔABC divides it into two triangles of equal areas.
ar(ΔABD) = ar(ΔACD)

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CBSE Class 9 Maths Chapter 9 Notes Quadrilaterals

Quadrilaterals Class 9 Notes Understanding the Lesson

Quadrilateral
A plane figure bounded by four line segments is called quadrilateral.

Properties:

• It has four sides.
• It has four vertices or comers.
• It has two diagonals.
• The sum of four interior angles is equal to 360°.

In quadrilateral ABCD, AB, BC, CD and DA are sides; AC and BD are diagonals and
∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.

Types of Quadrilaterals
1. Parallelogram: A quadrilateral whose each pair of opposite sides are parallel.

• AB || DC
• AD || BC

2. Rectangle: A parallelogram whose one angle is 90°. Diagonals are equal.

3. Rhombus: A parallelogram whose adjacent sides are equal.
Note: Diagonal bisect each other at 90°.

4. Square: A rectangle whose adjacent sides are equal (four sides are equal). Diagonal bisect each other at 90°.

5. Trapezium: A quadrilateral whose one pair of opposite sides are parallel. AB || DC

6. Kite: It has two pair of adjacent sides that are equal in length but opposite sides are unequal.

Note:

• One of the diagonal bisects the other at right angle.
• One pair of opposite angles are equal.

Properties of a Parallelogram

• Opposite sides are equal.
  e.g., AB = DC and AD = BC
• Consecutive angles are supplementary.
  e.g., ∠A + ∠D = 180°
• Diagonals of parallelogram bisect each other.
• Diagonal divide it into two congruent triangles. A B

Theorem 8.1: A diagonal of a parallelogram divides it into two congruent triangles.
Theorem 8,2: In a parallelogram, opposite sides are equal.
Theorem 8.3: If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
Theorem 8.4: In a parallelogram, opposite angles are equal.
Theorem 8.5: If in a quadrilateral, each pair of opposite angles of a quadrilateral is equal then it is a parallelogram.
Theorem 8.6: The diagonals of a parallelogram bisect each other.
Theorem 8.7: If the diagonals of quadrilateral bisect each other, then it is a parallelogram.
Theorem 8.8: A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.

Mid-point Theorem
Theorem 8.9: The line segment joining the mid-points of two sides of a triangle is parallel to the third.
Given: A triangle ABC, E and F are mid-points of sides AB and AC respectively.
i.e., AE = EB and AF = FC
To Prove:
(i) EF || BC
(ii) EF = \(\frac{1}{2}\) BC
Construction: Draw a line through C parallel to AB and extend EF which intersect at D.

Proof: (i) In AAEF and ACDF,
AF = CF (F is the mid-point of AC)
∠AFE = ∠CFD (Vertically opposite angles)
∠EAF = ∠DCF (Alternate interior angles)
∴ ΔAEF = ΔCDF (by ASA congruency)
∴ AE = CD (by CPCT)
and BE = CD (AE = BE)
EF = FD (by CPCT);
Hence, BCDE is a parallelogram.
ED || BC )
∴ EF || BC

(ii) BCDE is a parallelogram.
DE = BC
EF + FD = BC
2EF = BC
EF=\(\frac{1}{2}\)BC

Converse of Mid-Point Theorem

Theorem 8.10: The line drawn through the mid-point of one side of a triangle, parallel to another side  bisects the third side. ‘
Given: ΔABC in which E is the mid point of AB.
EF || BC
To Prove: AF = FC
Construction: Draw CD || AB and extend EF which intersect at D.
Proof: EF || BC
∴ ED || BC
AB || CD
⇒ BE || CD
∴ BCDE is a parallelogram.

Now in ΔAEF and ΔCDF, ∠AFE = ∠CFD (Vertically opposite angles)
∠EAF = ∠DCF (Alternate interior angles)
AE = CD (BE = AE opposite side of a parallelogram and BE = CD
∴ AAEF ≅ ACDF (by AAS congruency)
Hence AF = FC (by CPCT)

On this page, you will find Linear Equations in Two Variables Class 9 Notes Maths Chapter 8 Pdf free download. CBSE NCERT Class 9 Maths Notes Chapter 8 Linear Equations in Two Variables will seemingly help them to revise the important concepts in less time.

CBSE Class 9 Maths Chapter 8 Notes Linear Equations in Two Variables

Linear Equations in Two Variables Class 9 Notes Understanding the Lesson

1. Equation: An equation is a mathematical statement that two things are equal. It consists of two expressions one on each side of an equals sign. For example,7x + 9 = 0

2. An equation in a statement of an equality containing one or more variables.
7x + 3y = 10

3. Linear equation in one variable: A linear equation or first degree equation, in the single variable x is an equation that can be written in the form ax + b = 0 where a, and b are equal numbers, when a≠0.
Examples:

• 2x+3=0
• 3y + 4 = \(\frac{y}{3}\)
• 7x-\(\frac{9}{2}\) =0
• 3x -7y = 73

These equations are solved by applying the properties of real numbers and properties of equality.

4. Solution of a linear equation: The value of the variable which when substituted in place of variable makes both sides of the given equation equal, is called the solution of given equation. These values of variables is also known as root of the equation.
Example:
3x + 4y – 5
Let x – 3, and y = -1
Putting x- 3 and y = -1 in the given equation 3 x 3 + 4 x (-1) = 5
⇒ 9 – 4 = 5
⇒ 5 = 5
∴ LHS = RHS
Hence (3, -1) is a solution of given equation.

5. Linear equation in two variables: A linear equation in two variables is a first degree equation which can be written in the form ax + by + c – 0 and a, b both are non-zero real number. Where a, b and c are real numbers.
Examples:

• 3x + 2y – 9 = 0
• 7x – 4y + 6 = 0

6. Graph of a Linear Equation in two Variables
Graph of a linear equation in two variables is a straight line.

Steps of graphing a line

• If the equation is not in slope intercept form, i.e., y = mx + c, then write the equation in such form.
• Plot they intercept at (0, 6).
• Plot two or three more points by counting the rise and run from the y intercepts.

While solving the equation we should put the following points in our mind.                         •

• We should add or subtract the same number on both the sides of the equation.
• We should multiply or divide by the same non-zero real number on both sides of the equation.

Note:

• A linear equation in one variable has only one solution.
• A linear equation in two variables has infinitely many solutions.

(a) If the slope is positive, count upward for the rise and to the right for the run (also down and left)
Example;    y = \(\frac{2}{3}\) x + 1

(b) If the slope is negative, count downward for the rise and to the right for the run (also up and left)                  Example:    y =\(\frac{2}{3} \)x + 1

7. Draw a line through the points and place arrows on the ends. Extend the line to cover the whole grid (not just connect the two points)

Note:

• The graph of every first degree equation in two variables is a straight line.
• Equation of x-axis is y = 0 (:Hi) Equation of y-axis is x = 0
• The graph of x = a is a straight line parallel to y-axis.
• The graph of y = b is a straight line parallel to x-axis.
• Graph of the equation y = mx (i.e., has no intercepts) is a straight line passing through origin.
  Every point which lies on the graph of the linear equation in two variables is a solution of linear equation.
• Graph of linear equation in one variable
• If given equation is in variable x only then its value represented graphically is on x-axis.
• If the given equation is in variable y only then its value represented graphically is on y-axis.

8. Graph of linear equation in one variable

• If given equation is in variable x only then its value represented graphically is on x-axis.
• If the given equation is in variable y only then its value represented graphically is on y-axis.

For example, 2x = 5 ⇒ x=\(\frac{5}{2}\)
Representation: In one variable,

In Cartesian plane or in two variables,

Draw a line through \(x=\frac{5}{2}\) parallel to y-axis. In such representation, the equation has many solutions.