Circles Grade-9 ncert

circles definition

The collection of all points in plane,which are at fixed distance from a fixed points in the plane, is called circle. And the the fixed point is called center of the circle and the fixed distance is called the radius of the circle. Note that the line segment joint the center and any points on the circle is also called the radius of the circle.

The circle have 3 parts. One is the outside the circle is called exterior of the circle. Second  in side the circle which is called interior of the circle. Third is the circle itself. The circle and its interior will make up the circular region. if we take 2 points on the circle, P and Q, then the line segment PQ is called chord of the circle. The chord which is passing through the center of the circle is called diameter of the circle. The diameter is the biggest chord in the circle. And all the diameter in the circle is same. Also the diameter is the 2 times of the radius of the circle. We can also see that we can only draw maximum 4 numbers of diameter in a circle.

A piece of the circle between 2 points on the circle is called arc of the circle. The arc of the circle can be 2 types. 1 is minor arc and major arc, depending upon there length. If a circle have 2 same arc in there length , then it will called semi circle. The length of the complete circle is called the circumference of the circle. The region between a chord and either of its arc is called segment of the circle. Segment can be either major  or minor segment, depending upon there arc length as shown in the fig. the region between an arc and two radii joining the center to the two end points of the arc is called sector of the circle. It can also be Major sector and minor sector depending upon the arc length as shown in the fig.

See p-171 for the exercise 10.1

Angle subtended by a chord at a point

Consider the line segment PQ as the chord of the circle as shown in the figure and the point R on the circle. Then the angle PRQ is called the angle subtended by the chord PQ at the point R. The angle POQ is the angle subtended by the chord PQ at the center of the circle. We can also notice the major and minor arc PQ in fig.

Now we can examine the relation between the length of the chord and angle subtended by them at the center.

In the above fig. line segment AB, CD, EF are the equal chords of the circle. Now consider the triangle ABO and CDO . The sides AB and CD are same (given), side AO and CO are also same as both are radius of the same circle and the side BO and DO are also same as they are the radius of the same circle. So we can understand that triangle ABO and triangle CDO are congruent triangle (sss congruent rule). There for the angle AOB and COD are same (CPCT). So we can state this observation as a theorem.

Theorem 10.1. Equal chord of a circle subtend equal angle at the center.

Theorem 10.2 is just converse of the 10.1

Theorem 10.2 . If the angles subtended by the chords of a circleat the center are equal, then the chords are equal.

This one can also prove with the same figure. Consider the triangle ABO and CDO and take the angles AOB and DOC are same. Now we can say that the side OA and side DO are the same because they are the radius of the same circle. Same way we can say that side OC and OB are same as it also the radius of the same circle. Now we can say both triangle are congruent  by S(AS). There for the side AB and CD are same (CPCT). See p-173 for the exercise 10.2

perpendicular form the center to a chord

As shown in the figure consider the chord AB and the perpendicular OM. Now we can join the OA and OB, so that we will get two right angle triangle. We can see that side OA and OB are same, as it is the radius of the same circle. And the side OM is common for both triangle. Now we can say that the both triangle formed is a congruent triangles (RHS). Ther for we can say that AM =BM. And it can be conclude as a theorem

Theorem 10.3 The perpendicular from the center of a circle to a chord bisects the chord.

The covers of this theorem is

Theorem 10.4 The line drawn through the center of a circle to bisect a chord is perpendicular to the chord.

we can prove this by the following figure.

Here AM=BM (Given), OA = OB 9 (radius), OM=OM (cmone side). So all three sides of this triangle AOM and MOB are same, and these triangle are congruent by (sss). There for angle AMO = BMO=90° . Since the linear angle AB IS 180. So OM⊥ AB.

Perimeter and Area grade-7 Ncert

perimeter and area of rectangle and SQUARE

Perimeter is the total distance around the oundery. There for the the perimeter of the rectangle is l+b+l+b. or 2l+2b or 2(l+b).

Perimeter of rectangle = 2(l+b).

Area is the full space occupied in the rectangle. Ther for the area of the rectangle is l*b.

Area of the rectangle = l*b

Perimeter of the square is as same as rectangle. But the square have all the side same. So perimeter= Side+Side+Side+Side = 4*Side

perimeter of the square = 4*Side

Area of the square = Side*Side

Remember , if we cut some small rectangle from a big rectangle the perimeter of the big rectangle will increase and the area will decrease. see p-206. See p-208 for exercise 11.1

Triangle as the PArt of rectangle-11.2.1

Consider a rectangle ABCD with length 8 cm and width 5cm. And if we draw a diagonal of the rectangle it will make that rectangle in two equal right angle triangle. The area of the rectangle is 8*5=40 sqcm. As  this rectangle have two equal triangle , the area of the triangle will be the half of the rectangle. From this we can understand that the area of the triangle can be  found by ½*b*h where b is the length and h s the height of the rectangle. There for

Area of triangle = ½*b*h.

Congruent parts of the rectangle

If a rectangle have two congruent part, then the area of each part will be half of the rectangle.

And also we can say that  if a rectangle have  n number of congruent parts then each part have the area same as the area of the rectangle/ n. see p-210.

area of a parallelogram
Area of Parallelogram

Handling Data grade-5 NCERT

bar graph

Above figure shows the bar graph. see p-221. The bar graph have the horizontal scale and vertical scale. In this graph the horizontal scale contain the number of the baggage filed. And the vertical scale contain the kinds of the trash bag collected (included in the graph).

Each square of the graph shows the number of baggage collected. We can also find out the number of each kind of the baggage. And can be compare the each kind. And can also find the total baggage in all kind.

circle graph

Circle graph shows the data in different part of the  circle. and the whole part together will form a circle. In this the total number of items/kinds will consider as the whole. And the number of  each item will decide the size of the each part of the circle. In the circle graph the information is in the form of fractional number. The denominator represent the whole part and the numerator represent the number/quantity of that particular item. The below figures shows the data of the energy sources of 24 towns.

Pleases do the try this in the figure. S p-231 for the exercise 15a.

Tally marks

In this method of data handling, one small standing ling is used fr 1, 2 small standing line for 2. And each 5 standing line wile make as a group by cutting the 5 standing line in with a slanting line.  This is called tally mark. See p-224 for  more details. and p-225 for exercise 15b.

line graph

In this data1 will be marked o horizontally ii paragraph paper. And the corresponding  data2 will be in vertically. The bellow graph shows the years in horizontally and the corresponding temp. in vertical in a graph paper.

From this type of chart we can easy under stand the following things.


see p-227,228 foe exercise. 15c. chapter check-up p-229.

More about Decimals grade-5 NCERT

Multiplication of decimals

we can multiply the decimals by adding them ex. 3×0.7 is 0.7+0.7+0.7=2.1

see the models in page no. 116.

multiplying with out model

ex. 1.25×7

multiply as a whole number and ignore the decimals. Count the total numbers of the decimals places in both numbers.from right cont the same number of decimal places and put the point 125×7=875 after put the point we will get 8.75.

zeros in the products

ex. 0.02×3 as whole number we will get 6. but the total decimal place is 2 so we can add one more zero in the left side of 6 and can make 2 decimal place. 0.06.

ex2. 0.05×6 as a whole number we will get 30. after punting the decimal place we will get 0.30. but remember the the zeros of the end of the decimal number have no values. so it will become 0.3.

see common mistakes in p-117.

Multiplication by 10, 100, 1000

multiplying by 10 move the decimal point one place to right.

multiplying by 100 move the decimal point two place to right.

multiplying by 1000 move the decimal point three place to right.

ex 10×5.623=56.23

ex2 100×5.623=562.3

ex3 1000×5.623=5623

see p-118 for exercise 8A

Dividing decimal with whole numbers

ex1. 0.8/4 which means divide 0.8 in equal 4 parts. models can be see  in p-119.0.8/4=0.2

ex.2 p-119. 1.33/7 we can divide as whole number and put the same decimal place in the dividend to the quotient. So 1.33/7=0.19 we can check the answer by dividend= quotient x divisor.

divisor smaller than dividend

ex 4.35/3 this one can also divide as whole number. and put the decimal place as same number in the dividend. So 4.35/3=1.45.

reminders while dividing decimals

If we get reminder while dividing the decimals, then put the decimal point in the quotient as same as the dividend. Once the quotient have the decimal point we can add zeros in the dividend and can complete the division.

ex. 3.5/2 = 1.75

see p-120 for exercise 8b

division by 10.100,1000

To divide the decimal number with 10, 100, 1000 we can move the decimal point to left as much zeros in in the divisor.

ex. 321.5/10=32.15, 321.5/100=3.215, 321.5/1000=0.3215.

see p-121 for exercise 8c.

decimals and money

1 Rs is 100 paise. the model can be see in p-122. 50 paise can be write 50/100 rupee. that s 0.50 rupee.

So 1rupee+50 paise can be written as 1+50/100=1×50/100=1.50 rupee.

75 paise = 75/100= 0.75 rupee.

multiplication and division with money

Unitary method: In this method we can find the unit price of an item.

ex. A shop have he price for 8 lemon is Rs 20.  how can we find the price for 1 lemon. price for 1 lemon is 20/8. we can divide the rupees as same as whole number. but keep 2 decimal places in the quotient. Ie. 20/8 we will get 2.5. but we make it 2.50, to keep 2 decimal places in the rupee division. if the other shop have Rs 13.50 for 6 lemon, how we can compere the price for both shop. For that we have to find the price for the 1 lemon in each shop by dividing. After that we can compare the price for each shop. 13.50/6=2.25 a(We can divide by keeping decimal point also). The second shop have the low price. See more ex. in p-123. multiplication  can do as same as whole number. See p-124 for exercise 8D. The above method is find the price of the unit, or unit price so it is called unitary method.

See chapter checkup in p-125,126.



Area of a parallelograms and triangle grade-9 -01


If we have a triangular field and want to divide it in to 3 equal parts. Can we divide it with out finding the area. Suppose we divide only one side of the triangular field in three equal parts and joint the points to the opposite vertex.  is it equal in size?. We are handling this type of situation in this chapter. to check this we have to find the area of the each plane. The area of the plane is defined as the magnitude of  the planar region. And the planar region is the plane enclosed by the simple closed figure. We are also know about the congruent of figures. Two figures are congruent if they have in same shape and same size. then we can keep one figure over other. And if two figure are congruent then they are in same size. But the converse of the statement is not possible. Area of a figure can be write as some number of unit ex 12 cm sq.

In the figure 9.3-p-153. The figure T formed by two planar region formed by figure P and Q. wecan also denote the area of the figure A as ar(A) . The properties of the area of the figure is as follows

  1. If A and B are two congruent figures then ar(A) = ar(B)
  2. If a planar region is formed by a figure T is  made up of two non-overlapping planar regions formed by figure P and Q then ar(T) = ar(P) + ar(Q).
Figures on same base and between same parallel

See figure 9.4 and 9.5 on p-154.

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

see figure 9.6 and 9.7 on p-155 see exercise also.

parallelograms on the same base and between the same parallels

please see the activity 1and activity 2 in page 156/157. From this we can understand that, the parallelogram on the same base and between the same parallel are equal in area.

Theorem : 9.1parallelogram on the same base and between the same parallels are equal in area.

Proof: See page number 157 for details . See page 158,159 for example and exercise.


Fraction grade-5-01


fraction is used to represent the part of the whole thing. Fraction can be write in the form of   (Numerator /denominator).

ex. ½, ¾, ¼.

½ Means, we are cutting an apple in 2 equal part and taking only one part. The denominator will mention the whole thing is cutted in how much equal part. Here the apple is cutted in 2 equal part so the denominator become 2. And we are only taking one pice from that. So the numerator become 1 in this example. ½ can also be called as half.

Like fractions:  fractions with same denominator is called like fractions. ex. 3/7, 5/7, 1/7.

Unlike fractions : Fractions with different denominator is called unlike fraction ex. 4/7, 5/10, 5/9.

Proper fraction : if the numerator less than denominator in a fraction, then it is called proper fraction. The value of proper fraction is always less than 1. ex. 1/8, 4/9,6/11.

Improper fraction : If the numerator is equal to or greater than denominator, then it is called improper fraction. The value of improper fraction is always equal or greater than 1. ex. 5/2, 12/7, 8/8.

Mixed fractions:   This is the combination of a whole number and fraction. For example 3 apple and a half apple. ex 3½.

Comparing the like fraction : In Order to compare the like fractions we can just compare the numerator only and can decide which one is smaller, bigger or equal.  Do some exercise. p-79

Changing in to mixed fractions : we can change an improper fraction into mixed fraction by divide the numerator with denominator. The whole part will be the quotient, numerator will be the remainder and divisor will be the denominator of the mixed fraction. do exercise in p-79.

Changing in to Improper fractions : We can change a mixed fraction into improper fraction. The numerator of the improper fraction can obtain by multiplying the whole part and denominator and add the numerator with the product. The denominator of the improper fraction is same as the denominator of the mixed fraction see exercise in p-79. See video

Equivalent fractions

Same fraction of a whole can represent with different fractional numbers. that means different fractional number can be same value. ex. 1/2 of an apple and 2/4 an apple. See activity on p-80. we can find the number of equivalent fraction of a fractional number by multiplying the numerator and denominator by same number. We can also find the equivalent fractions by dividing the numerator and denominator by same number. see p-80

checking the equivalent fraction 

In order to check whether  to fraction are same or not we can cross multiply them. Cross multiplication can be done by multiply the numerator of the first number and denominator of the second number. Then multiply the numerator of the second number and denominator of the first number. If both product are same then the two fractions are same. The product is the value of the fraction whose denominator is multiplied. By this we can find which one is big or small also. see p-81 for exercise

lowest form of the fraction

As we know same fraction can be write with different numerator and different denominator. ex 2/4, 4/8,  6/ 12. So in this section we will study how we can check whether the given fraction is in its lowest form. If the given fraction number is in its lowest form then there will not be any common factor of numerator and denominator expect 1. (or the hcf will be 1).  For ex. 6/12, the HCF of 6 na12 is 6so it is not 1. There for 6/12 is not in its lowest form. If we consider 1/6, then the HCF is 1 so it is in it lowest form.

To change a fraction in its lowest form, we can find the HCF of numerator and denominator and then we have to divide both numerator and denominator with HCF  and we will get the lowest form of the fraction. ex 6/12 HCF is 6 so divide both numerator and denominator we will get 1/6 this is the lowest form. see exercise 6B in p-83. See video

Comparing the fractions

Comparing the like fractions:  as the denominator of the like fractions are same just compare the numerator. The biggest numerator fraction is the bigger one.

comparing the unlike fraction : In this case all denominator will be different. In this case if the numerator is same for all  the biggest denominator fraction is the smallest one. If denominator and numerator are different for fractions then we can find the LCM if the denominators and make all fraction with same denominator as LCM, then we can compare the numerator only. To do this we have to divide the LCM with denominator of each fraction. And multiply  the quotient with numerator and denominator of each fraction respectively. See exercise 6c p-86 See video

Adding the fractions

Adding the like fractions : As the like fraction have the same denominator we have to add only the numerators of the fractions and the denominator is the same.ex. 2/12 + 4/12 = 6/12.

Adding the unlike fractions: In order to add two or more unlike fractions we have to make them  like fractions. To do this we have to find the LCM of the all denominators, and convert the denominator of all the fractions to the LCM. Now all the fraction is in like fractions and we can add them. see examples and exercise in page 88. See video

Adding the mixed numbers with like fractions : For do this we have to add the whole part together first, and this will be the new whole part. Now add the two like fraction part. If the result is in improper fraction convert that in to mixed fraction and now we will get the proper fraction in the fraction part of the mixed fraction. Now we can add the previously obtained new whole part along with the whole part of the mixed fraction and keep the fraction part as it is. See ex. i page-89.

Adding the mixed numbers with unlike fractions : This one also we can do as per the above steps. But before doing that we have to covert the unlike fractions in like fractions. see ex.  and exercise in page-89.

Subtracting the fractions –

See video

To subtract any faction, we have to make them first in to the like fraction. To do this we can find  LCM of all denominators, and make all the fractions denominator to the LCM. Then we can subtract the numerators and keep the denominator same. Like this we can subtract like fractions and unlike fractions. even it is proper or improper. see the exercise i page-90.

Subtracting the mixed fractions:

Heron’s formula grade 9- NCERT-01


We all ready know about how to find out the area, perimeter of different shapes like rectangle and square. In this chapter we will study details about how we can find the area of various type of triangle.

area of the triangle

Area of the triangle = ½ * base * height

We can see that if the triangle is right angle triangle, we can directly take two side, which contain the right angle, as base and height. see the the fig.12.1 in p-197. So in this triangle ABC, BC and CA are the height and base ( any one can select as base or height).  There for the

Area of the triangle ABC = ½ * 5 * 12  =30 cm².

Now we can consider an equilateral triangle PQR as shown in fig 12.2 p-198. In this we can know all the side of the triangle. But how we can find the height. We can find the midpoint of QR and mark as M and join it to P. Now we can see that ΔPMQ is aright angle triangle. And we can find the PQ using the pythagorean theorem.

PQ ² = PM² + QM²


PM²= 75

PM= √75=5√3

Area of the triangle = ½ * 10 * 5√3 =25√3 cm².

Now we can consider an isosceles triangle. In this two sides is same. How we can find the height. We can consider the triangle XYZ in the fig 12.3 p-198. Find and mark the midpoint P on YZ and joint it with point x. Now we can see the triangle XYP ia right angle triangle.  So

XP² = XY² – YP² = 5² -4² = 25-16=9

XP= √9 = 3

Area of the triangle = ½ * 8 * 3 =12 cm².

Now we can consider the scalene triangle. In this type of triangle all sides are different. So we cannot find the height of the triangle as described in the above method. Her we can use the Heron’s formula. Heron is the famous mathematician (p-199). The formula given by him can use to find the area of any type of triangle. And it is called hero’s  formula

Area of a triangle = √(s(s-a)(s-b)(s-c))

Where a,b,c are the three side of the triangle. ANd s is the semi perimeter of the triangle. So  s= (a+b+c)/2.

Now consider the triangle ABC in the fig 12.5 p-200. Here a= 40, b=24,c=32. so

s= (40+24+32)/2=  48

s-a= 48-40=8



Area of the triangle = √(48*8*24*16) =384 m² see page 200,201,203 for exercise

we can also use this formula for finding the area of quadrilateral by dividing it in to two triangle, see page 206 for exercises.


Triangle and its property grade-7 NCERT-1

Triangle is a simple closed curve made up with 3 lines segment. it have 3 vertex, 3 side and 3 angle. In a triangle ABC, the side are AB, BC, CA. Angles are ∠BAC, ∠ABC, ∠BCA. Vertices  are A,B,C. In the above triangle the side opposite to the vertex A is BC. please try to say the opposite side of other vertex.

Based on side triangle can classifieds in to Scalene ( all three sides are unequal), Isosceles (2 sides are same), And Equilateral triangle (3 side are same).

Based on Angles Triangle can classifieds in to Acute-angle, Obtuse-angle, and Right angle triangle. see page 113.

Medians of triangle

A median is the line segment between a vertex and the midpoint of the opposite side. A triangle can be 3 medians. If we cut a triangle in a sheet of paper we can fold each of its sides and can be find the midpoint of each side.  see (P114) for think and discuss.

Altitude of a triangle

It is nothing but the height of the triangle. It is the line segment between the base and opposite vertex. See (P115) for think and discuss. See (P116) for exercise.

LCHF diet plan, food which can be eat or not eat

Food can be eat

Egg, All fish, Read meat, chicken, Olive Oil, Coconut oil, Broccoli, Cauliflower, Mushrooms, cabbage, tomato, ladies finger, curry leaf, coriander leaf, Green leafs vegetable, chilli, Bitter Gourd, Full fat milk and yogurt, butter, cheese, Dry fruits like, almond, walnut, pumpkin seeds. Drink at least 3.5 liters water, use lemon water with salt. Coconut

food CANNOT eat

Sugar, sugar product, juses , fresh fruits, Rice, chapati , perotas, beverages, bakery items. Potato, carrot, all items growing under the ground  . dal, and all type payer items,