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CIRCLES - Class IX, NCERT (CBSE) Mathematics Textbook Exercise 10.3 Solutions
Class IX, NCERT (CBSE) Mathematics Textbook
Chapter 10, Circles
Exercise 10.3 (Solution)
(Page 176)
Q 1: Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?
Ans 1:
Consider the following pair of circles.
The above circles do not intersect each other at any point. Therefore, they do not have any point in common.
The above circles touch each other at 1 point X only. Therefore, the circles have 1 point in common.
These circles intersect each other at two points G and H. Therefore, the circles have two points in common. It can be observed that there can be a maximum of 2 points in common. Consider the situation in which two congruent circles are superimposed on each other. This situation can be referred to as if we are drawing the circle two times.
Q 2: Suppose you are given a circle. Give a construction to find its centre.
Ans 2:
The below given steps will be followed to find the centre of the given circle.
Step1. Take the given circle.
Step2. Take any two different chords AB and CD of this circle and draw perpendicular bisectors of these chords.
Step3. Let these perpendicular bisectors meet at point O. Hence, O is the centre of the given circle.
Q 3: If two circles intersect at two points, then prove that their centers lie on the perpendicular bisector of the common chord.
Ans 3:
Consider two circles centered at point O and O’, intersecting each other at point A and B respectively.
Join AB. AB is the chord of the circle centered at O. Therefore, perpendicular bisector of AB will pass through O.
Again, AB is also the chord of the circle centered at O’. Therefore, perpendicular bisector of AB will also pass through O’.
Clearly, the centers of these circles lie on the perpendicular bisector of the common chord.
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Class IX, CIRCLES (Exercise 10.2) | NCERT (CBSE) Mathematics
Chapter 10, Circles
CBSE Class IX Mathematics
NCERT Solutions of Math Textbook Exercise 10.2
NCERT Solutions of Math Textbook Exercise 10.2
(page 173)
Q 1: Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centers.
Ans 1:
A 'Circle' is an array of points which are equidistant from a fixed point. This fixed point is called as the 'Centre' of the circle and this equal distance is the 'Radius' of the circle. Therefore, if we try to superimpose two circles of equal radius, then both circles will cover each other. Therefore, two circles can be congruent only if, they have equal radii.
Consider two congruent circles having centre O and O' and two chords AB and CD of equal lengths.
In ΔAOB and ΔCO'D, we have -
AB = CD (Chords of same length)
OA = O'C (Radii of congruent circles)
OB = O'D (Radii of congruent circles)
Hence, ΔAOB ≅ ΔCO'D (by SSS congruence rule)
∠AOB = ∠CO'D (By CPCT)
Hence, it is proved that equal chords of congruent circles subtend equal angles at their centers.
Ans 2:
Say, there are two congruent circles (circles of same radius) with centers as O and O' as shown in the following figure
In ΔAOB and ΔCO'D, we have -
∠AOB = ∠CO'D (Given)
OA = O'C (Radii of congruent circles)
OB = O'D (Radii of congruent circles)
Hence, ΔAOB ≅ ΔCO'D (by SSS congruence rule)
AB = CD (By CPCT)
Hence this is proved that if chords of congruent circles subtend equal angles at their centers, then the chords are equal.
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Class 9, NCERT (CBSE) Mathematics | Chapter 10, CIRCLES
Class IX, Mathematics (Circles)
NCERT (CBSE) Textbook Exercise 10.1 Solution
(Page 171)
Q1: Fill in the blanks
(i) The centre of a circle lies in __________ of the circle. (exterior/ interior)
(ii) A point, whose distance from the centre of a circle is greater than its radius lies in __________ of the circle. (exterior / interior)
(iii) The longest chord of a circle is a __________ of the circle.
(iv) An arc is a __________ when its ends are the ends of a diameter.
(v) Segment of a circle is the region between an arc and __________ of the circle.
(vi) A circle divides the plane, on which it lies, in __________ parts.
Ans 1:
(i) The centre of a circle lies in interior of the circle.
(ii) A point, whose distance from the centre of a circle is greater than its radius lies in exterior of the circle.
(iii) The longest chord of a circle is a diameter of the circle.
(iv) An arc is a semi-circle when its ends are the ends of a diameter.
(v) Segment of a circle is the region between an arc and chord of the circle.
(vi) A circle divides the plane, on which it lies, in three parts.
Q2: Write True or False. Give reasons for your answers.
(i) Line segment joining the centre to any point on the circle is a radius of the circle.
(ii) A circle has only finite number of equal chords.
(iii) If a circle is divided into three equal arcs, each is a major arc.
(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.
(v) Sector is the region between the chord and its corresponding arc.
(vi) A circle is a plane figure.
Ans 2:
(i) True. All the points on the circle are at equal distances from the centre of the circle, and this distance is called as radius of the circle.
(ii) False. There are infinite points on a circle and we can draw infinite number of chords of given length. Hence, a circle has infinite number of equal chords.
(iii) False. Because of each arc, the remaining arc will have greater length.
(iv) True. Twice the length of radius of any circle is its diameter.
(v) False by virtue of its definition.
(vi) True. A circle is always two-dimensional figure which is also a part of a plane.
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CLIMATE (Chapter 14,Contemporary India - I) | Class IX, NCERT (CBSE) Geography | Sample Objective Questions
Chapter-14, Contemporary India - I, CLIMATE
Class IX NCERT (CBSE) Social Science (Geography)
additional Questions-Answers
Q.1: Give reasons as to why,
(i) The bulk of rainfall in India is concentrated over a few months. [Textbook Q. No 4 ii]
(ii) The Tamilnadu coast receives winter rainfall. [Textbook Q. No 4 iii]
(iii) The delta region of the eastern coast is frequently struck by cyclones. [Textbook Q. No 4 iv]
(iv) Parts of Rajasthan, Gujarat and the leeward side of the Western Ghats are drought-prone. [Textbook Q. No 4 v]
Q.2: What is an ‘Equable Climate’? Give example.
Q.3: What is ‘Extreme Climate’? Give examples.
Q.4: What is ‘Rain Shadow’?
Q.5: What are the elements of Weather and Climate?
Q.6: Which place in India records the highest rainfall in the world?
Q.7: Which part of India experiences the highest range of temperature in a day?
Q.8: Name the season in which Eastern Coast (Coromondal) receives rainfall.
Q.9: What are Jet Streams?
Q.10: What are the reasons behind the disasters like flood, cyclones etc. along the coastal regions of Orissa and Andhra Pradesh?
Q.11: Why Chennai has more rains only after the fury of the monsoons is over in most parts of the country?
Q.12: Write short notes on:
(a) Loo
(b) Kaal Baisakhi
(c) Mango Shower
(d) Mahawat
Click here to see the Answers
CLIMATE (Contemporary India - I) | Class IX Geography | Answers of NCERT (CBSE) Textbook Exercise Questions [Read]
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CLIMATE | Class IX NCERT (CBSE) Geography | Contemporary India - I
NCERT / CBSE, Social Science (Geography)
Contemporary India - I
Chapter-14, CLIMATE
Textbook Exercise Questions - Answers
Q.1: Choose the correct answer from the four alternatives.
(i) Which one of the following places receives the highest rainfall in the world?
a. Silchar
b. Cherrapunji
c. Mawsynram
d. Guwahati
(ii) The wind blowing in the northern plains in summer is known as
a. Kaal Baisakhi
b. Trade winds
c. Loo
d. None of the above.
(iii) Which one of the following causes rainfall during winters in the North-Western part of India:
a. Cyclonic depression
b. Retreating monsoon
c. Western disturbances
d. Southwest monsoon
(iv) Monsoon arrives in India approximately in:
a. Early May
b. Early July
c. Early June
d. Early August
(v) Which one of the following characterizes the cold weather season in India?
a. Warm days and warm nights
b. Warm days and cold nights
c. Cool days and cold nights
d. Cold days and warm nights
Ans: (i) c (ii) c (iii) c (iv) b
Q.2: Answer the following questions briefly.
(i) What are the controls affecting the climate of India?
(ii) Why does India have a monsoon type of climate?
(iii) Which part of India does experience the highest diurnal range of temperature and why?
(iv) Which winds account for rainfall along the Malabar Coast?
(v) What are Jet streams and how do they affect the climate of India?
(vi) Define monsoons. What do you understand by “break” in monsoon?
(vii) Why is the monsoon considered a unifying bond?
Ans 2(i):
Various factors, both lying within and outside India influence the climate of India. The chief among them are the following -
(1) Latitude: The Tropic of Cancer passes through the middle of the country from the Rann of Kutchh in the west to Mizoram in the east. The areas lying in the south of the Tropic of Cancer belong to the tropical area while the remaining areas lying in the north of Tropic of Cancer belong to the sub-tropical area. Therefore, India’s climate has characteristics of tropical as well as sub-tropical areas.
(2) Altitude: With the increase in altitude the atmosphere becomes less dense and temperature decreases. That is why hills are cooler during summer. India has an average height of 6000 meters in the north while in the south there is a vast coastal area with maximum elevation of about 30 meters. It is because of the differences in altitudes or relief that while it is hot during summer in the Plains of Punjab, Haryana, UP etc., the places in the mountains like Nainital, Darjeeling, and Ooty etc. are cool.
(3) Physiography: Physiography also plays an important role in determining the climate of a place. India has a varied Physiography. The Himalayas in the north act as great barrier to the cold winds from Central Asia. In absence of the Himalayas, India would have been a very cold country. The location of the Arabian Sea, the Indian Ocean and the Bay of Bengal on the three sides of India exert a moderating influence on the climate of India. This ocean, sea etc. being a huge water body acts as a store house of moisture which India needs very badly. Without these water bodies India would have been a very hot country with a climate like that of a dessert like Sahara in Africa.
(4) Western Cyclonic Disturbances: The western cyclonic disturbances are weather phenomena of the winter months brought in by the westerly flow from the Mediterranean region. They usually influence the weather of the north-western regions of India. In winter, in northern India, Punjab and Haryana get some rain from these western disturbances.
(5) Pressure and Winds: India lies in the region of north-easterly winds. These winds originate from sub-tropical high pressure belt of northern hemisphere. The fury of monsoons as well as long dry spell in India is mainly due to the pressure conditions which develop in the surrounding countries like East Africa, Iran, and Central Asia.
Winds especially Typhoons originating in China Ocean currents along with onshore winds and monsoon winds have a great control on the climate of India. Also the upper air circulation such as Jet Streams play a dominant role in determining the climate of India.
Ans 2(ii):
The word “Monsoon” is derived from the Arabic word ‘Mausim’ which literally means season. Monsoon refers to the seasonal reversal in the wind direction during a year. The monsoon winds are confined to the tropical area roughly between 20ON to 20OS latitudes. But in the Indian subcontinents, because of the Himalayan ranges, they bring the whole subcontinent under the sway of the moist bearing winds for 2 - 5 months roughly between May to October. Almost 90% rainfall in India is due to these monsoon winds. India would have been an arid land or desert if there had been no phenomena of monsoons. It is because of these reasons that the climate of India is described as monsoon type.
Ans 2(iii):
The difference in temperature of a particular place in a single day is called the diurnal change of temperature. The highest diurnal range of temperature has been found in Thar Desert of Rajasthan. At this place the day temperature may rise to 50OC and may drop down to near freezing point the same night.
The highest diurnal range of temperature has been found at the Thar Desert because there the weather conditions drastically change during the day and during the same night. The great difference between the day and night temperatures result in the highest range of diurnal range of temperature in the Thar Desert.
Ans 2(iv):
South-West monsoons account for the rainfall along the Malabar Coast while North-East monsoons account for the rainfall along the Coromeodal Coast or East Coast.
Ans 2(v):
Jet Streams are fast flowing winds blowing in a narrow zone in the high altitude above 12000 m in troposphere. There are a number of separate jet streams whose speed varies from 110 km/h in summer to about 184 km/h in winter.
In winter the sub-tropical westerly jet streams bring rain to the western part of India, especially Himachal Pradesh, Haryana and Punjab. In summer the sub-tropical easterly jet blows over Peninsular India approximately at 14ON and bring some rain and storm.
Ans 2(vi):
Monsoon - The word “Monsoon” is derived from the Arabic word ‘Mausim’ which literally means season. Monsoon is thus the rain-bearing winds which reverse their direction with the change of ‘mausam’ or season and blow throughout the Indian subcontinent, especially from May to October.
Break Monsoon - The monsoon rain takes place only for a few days continuously at a time. They are inter-spread with rainless intervals. Thus monsoon rains have wet and dry spells. So, this phenomena associated with the rainfall during monsoon is known as ‘break’.
Ans 2(vii):
India is a vast country with varied topography. On the north the Himalayas protect the sub-continent from the extreme cold winds of Central Asia. This enables northern India to have uniformly higher temperatures as compared to other areas on the same latitudes. Similarly the peninsular plateau, under the influence of sea from three sides, has moderate temperatures. Despite such moderating influences there are great variations in the temperature conditions.
Nevertheless, the unifying influence of the monsoons on the Indian subcontinent is quite perceptible. The Indian landscape, its animal and plant life, the agriculture calendar and the life of the people including their festivities and economic conditions revolve around the monsoon. Year after year, the people of India from North to South and from East to West eagerly await the arrival of the monsoon. These monsoon winds bind the whole country by providing water to set the agricultural activities. Thus, in view of the above the monsoon is considered as a unifying bond for this subcontinent.
Q.3: Why does the rainfall decrease from the east to the west in Northern India?
Ans: In summer, the monsoons rise both from the Bay of Bengal and the Arabian Sea. The monsoons which rise from the Bay of Bengal cause heavy rainfall in the Brahmaputra Valley and the Assam Hills (i.e. in Cherrapunji). Because of the Himalayas they then take a western turn and move up the Ganga Valley but as they proceed westwards they become drier and drier and therefore they cause less and less rain as they move forward. As such Kolkata gets a rainfall of 119 cm, Patna 105 cm, Allahabad 76 cm and Delhi 56 cm.
Q.4: Give reasons as to why,
(i) Seasonal reversal of wind direction takes place over the Indian subcontinent?
(ii) The bulk of rainfall in India is concentrated over a few months.
(iii) The Tamilnadu coast receives winter rainfall.
(iv) The delta region of the eastern coast is frequently struck by cyclones.
(v) Parts of Rajasthan, Gujarat and the leeward side of the Western Ghats are drought-prone.
Ans 4(i): Monsoons blow from northeast India towards the sea during winter (November to April) but with the beginning of summer they begin to reverse their direction. During winter (October - November) with the apparent movement of sun towards the south, the low pressure trough over then northern plains becomes weaker which causes the monsoon wind to blow from northeast to south during this period.
In summer, areas of low pressure develop in north and northwest parts of India due to which winds from the high pressure area i.e. sea begin to blow towards the land masses. So, a shift in the development of monsoon trough or low pressure trough along with the change of season is the main reason for the reversal of wind direction in Indian subcontinent.
Answers of Q.4(ii) to 4(v) are given in a separate post. [Check the answers of Sample and Objective Questions, link given below].
Q.5: Describe the regional variations in the climatic conditions of India with help of suitable examples.
Ans: The climate of India is described as that of the monsoon type. But within this general pattern there are found certain regional variations in climatic conditions. This is because of the variations in temperature, precipitation, atmospheric pressure, wind, humidity and altitude from place to place. The following are a few examples which prove the above fact:
Range of Temperature - Temperature has great bearing on the climate, so difference in temperature is bound to create variation in the climate. In India there are places like, Rajasthan and south-west Punjab, where the mercury rises even up to 55OC. On the other hand, there are places like Dras, near Kargil, where the temperature sometimes, touches as low as -45OC.
Direction of the Rain-bearing Winds - The direction of the Rain-bearing winds has a great impact on the climate of a place. The summer monsoons arising from the Arabian Sea because of their south-west direction strike the Western Ghats first and cause a heavy rainfall there (about 250 cm). But these winds reach the Eastern Ghats last of all so there is less rainfall in Tamil Nadu and consequently it is much lower as compared to the rainfall on the Malabar Coast of the Western Ghats.
Form of Precipitation - The form of precipitation whether it is in the form of light rains or heavy snow has also a great-bearing on the climate of an area. In winter north-west of India gets some rains due to the Western Disturbances. As a result, there is little rain in the plains of Punjab and Haryana but there is heavy snowfall in the Western Himalayas especially in Himachal Pradesh and Jammu and Kashmir. It is all due to the change in the form of precipitation.
Amount of Rainfall - Difference in rainfall is bound to create variations in climate. In India, there are places like Mawsynram which receives 1080 cm of rainfall annually. This is perhaps the highest rainfall all over the world. On the other hand, there are places in India, especially in Rajasthan, which gets 20 cm of annual rainfall.
Rainfall Regime or Seasonal Distribution of Rains - In India, there are many parts which get rains only in summer while there are others which dry in that season. On the other hand, there are certain places which get rains in winter alone while there are others which get scanty or no rainfall in winter. For example Tamil Nadu and AP get much of their rainfall in winter season, and in summer they are almost dry. This seasonal distribution of rainfall has a great bearing on climate. In summer both Tamil Nadu and AP experience dry and hot season while the rest of the country especially Kerala, Karnataka and Maharashtra on the west-coast of India has a pleasant climate.
Q.6: Discuss the mechanisms of monsoons.
Ans: The word ‘monsoon’ refers to the seasonal reversal in the wind direction during a year. The monsoon winds are confined to the tropical area lying between 20ON to 20OS latitudes. The mechanism of monsoons or the phenomena of the seasonal reversal in wind direction is related to the following fact:
1. The differential heating and cooling of land and water.
2. The shift of the position of Inter Tropical Convergence Zone (ITCZ) in summer over the Ganga plain.
3. The presence of high pressure area, east of Madagascar over the Indian Ocean. The intensity and position of this high pressure area affects the Indian monsoon.
4. The Tibetan plateaus gets intensely heated during summer resulting into strong vertical air currents and the information of low pressure over the plateau.
5. The movement of the westerly Jet Stream to the north of the Himalayas and the presence of the tropical easterly Jet Stream over the Indian Peninsula during summer.
The meteorologists have found out a seesaw kind of relationship between the meteorological changes going on over the Pacific and the Indian Ocean. Whenever there is high pressure in the subtropical region of the Pacific Ocean in the northern hemisphere, the pressure in the south of Indian Ocean tends to be low and vice-versa. This is known as the ‘Southern Oscillation’ or SO which causes the shifting of winds across the equator in different directions. Besides the above facts, some other events such as ElNino, ENSO etc. are also connected to the mechanism of monsoons.
Q.7: Give an account of weather conditions and characteristics of the cold season.
Ans: The cold weather persists from December to February throughout India. January is the coldest month. During this period temperature varies from 10OC to 15OC in northern plains and 25OC in southern parts of the country. There is high pressure in northern plains due to the cold climatic conditions. The cold dry winds blow from the high pressure towards the low pressure equatorial regions. These winds are north-westerly in the Ganga plains and north-easterly over the Bay of Bengal. Only two parts of the country receive rain in this season. First is the north-west part of the country which receives rainfall, caused by the Western Cyclonic Disturbances originating in the Persian Gulf and Mediterranean regions. Second is the coast of Tamil Nadu which gets rainfall in winter because of the north-east winds on Retreating Monsoons which blow over the Bay of Bengal. But in other parts of the country, the weather is pleasant with clear skies and bright sunshine.
Q.8: Give the characteristics and effects of the monsoon rainfall in India.
Ans: Chief Characteristics and Effects of the Monsoons or Monsoon Rainfall
(a) The monsoons account for 90% of the rainfall received by India. Almost every part of the country except the east coast of Tamil Nadu receives the bulk of rainfall from the summer monsoons.
(b) The rainfall is neither continuous nor regular even in the wettest areas.
(c) Overall rainfall is not the same each year. The monsoons may be strong or weak in a particular year depending on the intensity and frequency of depression in the Bay of Bengal.
(d) The monsoon rains are not evenly distributed.
(e) A low pressure near the Himalayas may result in heavy rains in the hills causing floods in the plains or the plains may face prolonged dry spells.
(f) Sometimes, the monsoons may break too early or too late. There may be too long or too many breaks in the monsoons, which may retreat too early or too late. All these factors upset the agricultural operations making agriculture in India a mere gamble.
Sometimes, the effects of the monsoons or its uneven distribution create a number of problems. Like if there is too much rain it causes floods and havoc all around. On the other hand insufficient rain in a year or over some years results in draught, famine, food-insecurity, starvation and death.
Sample Questions, on this chapter
Climate (Chapter 14, Contemporary India - I) | Sample (Objective) Questions - Answers [Read]
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