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Showing posts with label 9-Maths. Show all posts
Showing posts with label 9-Maths. Show all posts

## Ncert solutions for Maths  class 9 --Circles

A circle is collection of all the points in a plane which are equidistant from a fixed point in the plane

Equal chords of a circle subtend equal angles at the centre

The angle subtended by the two chords of a circle at the centre are equal then the chords are equal

The perpendicular from the centre of a circle to a chord bisects the chord

Equal chords of a circle are equidistant from the centre and vice versa

Angles in the same segment of a circle are equal

Angle subtended by an Arc at the centre is double the angle subtended by it at any point on the remaining part of the circle

Angle in a semicircle is a right angle

The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degree

Sum of pair of opposite angles of a quadrilateral is 180 degree then the quadrilateral is cyclic

Ex 10.1

Que 1

Fill in blanks

1) The centre of the circle lies in the------ of the circle

Ans-- interior

2) A point whose distance from the centre of a circle is greater than its radius lies in the ------ of the circle

Ans Exterior

3) The longest chord of a circle is   ------ of the circle

Ans Diameter

4) An Arc is a----- when its ends are the ends of a diameter

Ans semicircle

5) Segment of a circle is the region between an Arc and ------ of the circle

Ans  chord

6) A circle divides the plane on which it lies in ----- parts

Ans --- three

Que 2 True and False

1) Line segment joining the centre to any point on the circle is a radius of the circle. True

2) A circle has only finite number of equal chords. False

3) If a circle is divided into three equal Arc each is a major arc. False

4) A chord of a circle which is twice as long as its radius is a diameter of a circle. True

5) Sector is the region between the chord and its corresponding arc False

6) A circle is a plane figure  True

Exercise  10.2

Que 1

Recall the two circles are congruent if they have the same radii prove that equal chords of a congruent circles subtend equal angles at their centres

Solution---

Que 2

Prove that if chords of a congruent circles substance equal angles at their centres then the chords are equal

Solution--

Exercise 10 .4

Question

How to find the length of the common chord
There are two circles of radius 3 cm and 5 cm they intersect each other the distance between their centre is 4 cm find the length of the common chord

Solution

Que 2.
If two equal chords of a circle intersect within the circle prove that the segments of one chord are equal to corresponding segments of other chord

Que 3
IF two equal  chords of a circle intersect within the circle prove that the line joining the point of intersection to the centre makes equal angles with the chord

Que 4

Line intersects two concentric circles with centre O at A,B,C and D  prove that AB=CD

Que 5

Three girls Reshma Salma and Mandeep are playing a game by standing on a circle of radius 5 metre drawn in a park throws a ball to the Salma Salma to Mandeep Mandeep to Reshma if the distance between Reshma and Salman and between Salman Mandeep is 6 m each  what is the distance between Reshma in Mandip?

Que 6

Circular park of radius 20m is situated in a colony three boys Ankur Syed and David all sitting at equal distance on its boundary each having a toy telephone in his hands to talk to each other find the length of a string of each phone?

Exercise 10.5

Que 2

A chord of a circle is equal to the radius of the circle find the angle subtended by the chord at a point on the minor ARC and also at a point on the major arc

Solution

Que 3

∆ PQR =100° where P,Q,R are points on a circle with centre O Find angle OPR

Solution

Que 1

A,B,C are three points on a circle with centre O such that angle BOC =30° and angle AOB = 60° If D is a point on the circle other than the arc ABC find angle ADC

Solution

Que 4

Angle ABC =69° ,angle ACB = 31° find angle BDC

Solution

Que 5

A,B,C and D are four points on a circle AC and BD intersect at a point E such that angle BEC =130° and angle ECD = 20° Find angle BAC

Solution

Que 6

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E If angle DBC =70° angle BAC is 30 ° find angle BCD further if AB = BC find angle ECD

Solution

Que 7 if diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral prove that it is a rectangle

Que 8

If the non parallel sides of a trapezium are equal prove that it is cyclic

Que 9

Two circles intersect at two points p and c through b two line segments are drawn to intersect the circles respectively prove that

£ACP=£QCD

Que 10

Circles are drawn taking two sides of a triangle is a diameter prove that the point of intersection of these circles lie on the third side

Que 11

ABC and ADC are the two right triangles with common hypotenuse AC  prove that

Que 12

Prove that a cyclic parallelogram is a rectangle

## Friday, October 22, 2021

### Mock Test Series

1) In a quadilateral three angles are in ratio 3:3:1 one of angle is 80' find other angles?

a) 120,120,40
b) 100,100,80
c) 110,110,60
d) 90,90,30

Ans (a)

2) Two adjacent angles of rhombus are 3x-40and 2x+20 the measurement of greater angle is

a) 160
b) 100
c)80
d) 120

3) Which of following pair of angles are opposite angles of cyclic quadilateral?

a)131,28
b) 95,55
c) 123,57
d) 64,52

4) In a quadilateral if diagonals bisect each other at right angles AB=BC=AD=6cm find length of CD

(a) 3 cm
(b) 6 cm
(c) 6.5 cm
d) 12 cm

5) Two consecutive angles of a parallelogram are in ratio 1:3 what is value of smaller angle ?

a) 50
b) 90
c)60
d)45

6) The diagonals of a rhombus are 12 and 16 cm find the length of side of rhombus?

a) 10
b) 12
c) 16
d) 8

7) In a quadilateral if diagonal are in equal in length it is

a) rhombus
b) square
c) rectangle
d) none

8) In a trapezium ABCD AB = 20 cm CD =10 cm AB|| CD P and Q are mid points of AC and BD find length of PQ
a) 10
b) 5
c)2
d)15

## NCERT Solutions  - Class 9 -Maths - Probability

Ncert exercise 15.1

Que 1
In a cricket match batsman hits a boundary 6 times out of 30 balls she plays find the probability that she did not a boundry

Que 2
1500 families with 2 children were selected randomly and the following data were recorded

No of girls.          2.            1.             0
No of families  475.         814.        211

Compute the probability of a family chosen at random

1) 2 girls

2) 1 girl

3) No girl

Watch the attached lecture to solve ex 15.1

### Mock test series

Que 1
In a sample survey of 640 people it was found that 400 people have secondary school certificate if a person is selected at random the probability that the person does not have such certificate

a) 0.375

b) 0.625

c) 0.725

d) 0.875

Ans (a)

Que 2
Cricket match a batsman hits a sixer 8 times out of 32 balls played the probability that a sixer is not hit in a ball

a) 0.75

b) 0.25

c) -0.25

d) 0.5

Ans (a)

Que 3
A die is thrown once in it  is noted then the probability that it is a prime number

a) 1/6

b) 2/6

c) 3/6

d) 4/6

Ans (c)

Que 4
In an experiment the sum of probabilities of all events is

a) 2.5

b) 1

c) 0

d) 3/8

Ans (b)

Que 5
Coin is tossed hundred times the head others 58 times and tail occurs 42 times that's per mental probability of getting a head is

a) 1/2

b) 21/50

c) 29/50

d) 42/58

Ans (c)

Que  6
Out of 200 students it is observed that 125 students like mathematics what is the probability that the students who do not like mathematics

a) 5/8

b) 7/8

c) 3/8

d) 1/2

Ans (c)

Que 7
Out of 35 students participating in a debate 10 are girls probability that a winner is a boy

a) 1/7

b) 3/7

c) 4/7

d) 5/7

Ans ( d)

Que 8
Indian soldiers were practicing for a shooting game one shoulder hit 9 times out of 10  chances find the probability of hitting the target

a) 2/10

b) 9/10

c) 1/10

d) 1/2

Ans (b)

Que  9
The probability of happening of an event is 45% the probability of an event is

a) 45

b) 4.5

c) 0.45

d) 0.045

Ans (c)

Que 10
Sum of probability of happening and not happening of an event is

a) 1

b) 2

c) 0

d) none of these

Ans (a)

## NCERT Solutions for Class 9 -Maths- Introduction to Euclid Geometry

Greek developed geometry in a systematic manner
Euclid Greek mathematician father of geometry introduce the method of proving geometrical results by using logical reasoning
He wrote 23 definitions in the book of the "elements 7 "

Point
It has neither length  nor breadth it is denoted by the find out on a paper it is represented by the letter A

Line segment
The portion of a line is called a segment of line what is the subset of the line that has length it has two endpoints it can be measured

Collinear points
3 or more points are collinear if they all lie in the same line

Concurrent lines
three or more than three lines are concurrent if there is a point which lies on all of them

Watch the lecture to understand different concepts

Parallel lines

Two lines are said to be parallel when they are not intersecting and they are  coplanar

Perpendicular lines

Two lines lie in the same plane are said to be perpendicular as they form a right angle

Line segment

Line segment is a part of line with two endpoints and two distinct points a and b are on the line then the part of this line with end points A and B is called a line segment AB

Distance from the centre to a point on the circle is called the radius of the circle

Euclid to find a point a line and a plane and definitions are not accepted by mathematician

Axioms and postulates are the assumptions which are universal truth they are not proved

Terms are the statements which are producing definitions exams and previously proved statements and deductive reasoning

Euclid postulates

Postulate 1--
A straight line may be drawn from any one point to any other point

Postulate 2 --

Terminated line can be produced indefinitely

Postulate 3 --

A circle can be drawn with any centre and any radius

Postulate 4 --

All right angles are equal to one another

Postulate 5 --

Straight line falling on the two straight lines make the interior angles on the same side of it taken together less than the two right angles then the two straight lines if produced indefinitely meet on that side on which the sum of angles is less than the two right angles

### Multiple choice questions

Que 1
A surface is that which has

b) length only

d) length and height

Ans (a)

Que 2

"Lines are parallel if they do not intersect is" stated in the form of

a) axiom

b) definition

c) postulate

d) proof

Ans ( b)

Que 3

Two planes intersect each other to form a

a) plane

b) point

c) straight line

d) angle

Ans ( c )

Que 4

Which of the following needs a proof

a) Axiom

b) Theorem

c) Postulate

d) Definition

Ans ( b)

Que 5

Euclid stated that all the right angles are equal to each other in the form of

a) Axiom

b) Theorem

c) Postulate

d) Definition

Ans (c )

Que 6

Euclid stated that if equals are subtracted from the equals the remainders are equals in the form of

a) Axiom

b) Theorem

c) Postulate

d) Definition

Ans ( a )

Que 7

Euclid divided his famous treatise "The elements" into chapters

a) 13

b) 12

c) 11

d) 10

Ans  (a )

Que 8

The total number of prepositions in "The Elements " are

a) 460

b) 465

c ) 13

d) 55

Ans ( b )

Que 9

It is known that if x+y = 10 then x+y+z =10+z Euclid axiom that illustrate this statement is

a) first axiom

b) second axiom

c) third axiom

d) fourth axiom

Ans. ( b )

Que 10

The three steps from solids to points are

a) Solids -Surfaces - Lines - Points

b ) Solids - Lines - Surfaces - Points

c) Lines - Points - Surfaces - Solids

d) Points - Lines - Surfaces - Solids

Ans (a)

## NCERT Solutions for Maths Class 9 - Construction

Mock Test series

Que 1
It is not possible to construct an angle with the help of ruler and compass

a) 37.5°

b) 40°

c) 22.5°

d) 67.5°

Ans ( b )

Que 2

With the help of ruler and compass it is not possible to construct an angle of

a) 35°

b) 20°

c) 37.5°

d) 47.5°

Ans (  b. )

Que 3

The construction of a triangle ABC given that  BC = 6 cm and angle B = 45° it is not possible when the difference of AB  and AC  is equal to

a) 6.9

b) 5.2

c) 5

d) 4

Ans  (a )

Que 4

The construction of a triangle ABC  given that BC = 3 cm and angle C = 60° is possible when the difference of AB  and AC  is equal to

a) 3.2

b) 3.1

c) 4.5

d ) 2.8

Ans ( d )

Exercise 11.1

Que 1
Construct an angle of 90 degree at the initial point of a given Ray and justify the construction

Que 2

Construct an angle of 45 degree at the initial point of a given Ray and justify the construction

Solution---

Que 3

How to construct angle of 60°,30°,15° ,22½°

How to draw bisector of any angle

Solution--

Que 4
Construct Angle  of 135°  and verify by measuring them with the protector

Solution --

Exercise 11.2

Que 4

If perimeter  sum of all three sides are  given then construct

Construct a triangle if its perimeter is 10.4 and two angles are 45 degree and 120 degree
Steps of construction

1) Draw a line segment X Y = 10.4 cm

2) Make an angle LXY = 45 ° and angle MYX = 120 °

3) Bisect angle LXY and angle MYX  and let these bisectors intersect at a point A

4) Draw A perpendicular bisector PQ and RS of AX and AY  respectively

5) Let PQ intersect XY at B and RS intersect XY at C Join AB and AC

Thus ∆ ABC is required triangle
Watch the lecture

Practice question--

Construct a triangle xyz where some of perimeter is 11 cm and angle Yis  30 degree and angle Zis 90 degree

Que 2

Construct a triangle PQR in which
QR =6 cm ,£Q=60° and PR-PQ= 2cm

How to draw a triangle when difference of two sides are given

Steps of construction

1) Draw the line QR

2 ) Make an angle XQR = 45° at point Q

3) cut the line segment QS = QP - PR  from OX

4) Join SR and draw the perpendicular bisector of SR say AB

5 ) Let bisector AB intersects QX at P and join PR
Thus ∆ PQR is required Triangle

How to draw a triangle when sum of two sides are given

Que 1

Construct a triangle ABC in which BC =7cm ,£B =75° and AB + AC =13 cm

Steps of construction

1) Draw a line Segment BC

2) At point B draw an arc of angle XBC

3) cut a line segment equal to AC + AB on BX

4) join DC

5) Make an angle DCY = angle BDC

6) Let CY intersect BX at A

Thus this is required triangle

If  want to take doubt session contact on whatapp number

## Linear Equations in two variables

### Que 1

Number of lines passing through a point

( 3, 4)

a) one

b) two

c) three

d) infinite

Ans (d)

Que 2

Any point on the x-axis is of the form

a) (x,x)

b) (x,y)

c). ( 0,x)

d). ( x,0)

Ans ( d )

Que 3

Any point on the y axis is of form

a) (y,y)

b) (x,y)

c) ( 0,y)

d ) (y,0)

Ans (c)

Que 4

Linear equation in two variables how many solutions it has

a) one

b) two

c) infinite

d) not possible

Ans ( c )

Que 5

Which of the following is a solution of the equation 4x+3y=16

a) (2,3)

b) (1,4)

c) ( 3,2)

d). (4,1)

Ans (b)

Que 6

Which of the following lines passes through

(1,2)

a) x+y=3

b) x-y=0

c) x+2y = 0

d) 2x+y= 0

Ans. (a)

Que 7

The equation x= 4 represent the line

a) parallel to x axis

b) parallel to y axis

c) passes through origin

d) perpendicular to y axis

Ans (b)

Que 8

If (2,0) is solution of linear equation 2x+3y=k then the value of k is

a) 4

b) 6

c) 5

d) 2

Ans (a)

Que 9

The graph of y =mx is a straight line

a) parallel to x axis

b) parallel to y axis

c) passing through origin

d) coincides with x axis

Ans ( c)

Que 10

The point of the form (a,a) always lies on

a) x axis

b) y axis

c) on the line y=m x

d) on line x+y=0

Ans ( c )

### Recorded Lectures

Exercise 4.1

Que 1
The cost of a notebook is twice the cost of a pen write a linear equation in two variables to represent this statement

Que 2
Indicate the values of a b and c in each case in the linear equations in the ax+by+c

1) 2x+3y=9.35-

2) x-y/5-10 =0

3) 3x+2=0

4) 5=2x

Solution

Watch  lecture in order to get solutions of exercise number 4.1 of CBSE board class 9th NCERT book

Solution

Exercise 4.2

Que
Which one of the following options is true and why

y=3x+5

a) unique solution

b) only two solutions

c) infinetly many solutions

Que
write four solutions for each of the following equations

1) 2x+y=7

2) πx+y = 9

3) x=4y

Que 3
Find the value of k if x= 2,y=1, is a solution

of the equation 2x+3y=k

Solution

### Exercise 4.3

Que
Draw the graph of the following linear equations in two variables

1) x+y=4

2) x-y=2

3) y=3x

4) 3=2x+y

Solution

Que
The taxi fare in a city is as follows for the first kilometre the fare is rupees 8 for the subsequent distance it is rupees 5 per kilometre taking the distance covered as X km and total fare is rupees why find the linear equation for information and draw its graph

Solution

Que
Here is a linear equation that converts Fahrenheit to Celsius
F=9/5 C+32

Draw the graph of the linear equation using Celsius for x-axis and Fahrenheit for y-axis

The temperature is 30 degree Celsius what is the temperature in Fahrenheit

The temperature is 95 degree fahrenheit what is the temperature in Celsius

Que

Yami and Fatima two student of class 9th together 22 rupees hundred was prime minister relief fund to help the earthquake victims write a linear equation which satisfies the state are you may take their contributions x and y and draw the graph of the same

Solution

Summary of. Lesson

And equation of the form  ax+by+c=0 where a b c are the real numbers such that a and b are not zero is called linear equation in two variables

Linear equation in two variable has infinite many solution

The graph of every linear equation in two variable is a straight line

X =0 is the equation of the y axis and
y =0  is the equation of the x-axis

The graph of x =  a straight line parallel to the y-axis

Graph of a y=a  straight line parallel to the x-axis

## NCERT Solutions for Maths Class 9 - Surface Areas and Volumes

### Mock Test Series

Que 1

The volume of cube is 3√3then total surface area of cube is

a) 6

b) √3

c) 18

d) 18√3

Ans ( c)

Que 2

The lateral surface area of a cube is 1600 cm^2  then its edge is

a) 15

b) 18

c) 20

d) 25

Ans ( c )

Que 3

Surface area of a room if its length breadth and height is 10 cm,10cm,5cm

a) 500

b ) 25

c) 400

d) 1000

Ans (c)

Que 4

The length of the longest pole that can be put in a room of dimension 10 m×10m×5m

a) 15

b) 16

c) 10

d) 12

Ans (a)

Que 5

A cuboid has dimension 6 cm  4 cm and 2 cm Number of cubes of 2cm side that can be cut from cuboid is

a) 18

b) 6

c) 10

d) none of these

Ans ( b )

Que 6

Any cylinder radius is doubled and height is half the curved surface area will be

a) halved

b) doubled

c) same

d) four times

Ans  ( c )

Que 7

The radii of two cylinders are in ratio  2:3 and their heights are in ratio 5:3  the ratio of their volumes is

a) 10:17

b) 20:27

c) 17:27

d) 20:37

Ans  (b )

Que 8

The radius of hemispherical balloon increases from 6 cm to 12 cm as air is being pumped into it the ratio of surface area of the balloon in the two cases

a) 1:4

b) 1:3

c) 2:3

d) 2:1

Ans ( a )

Que 9

What is total surface area of hemisphere of radius 10cm

a) 1000

b) 942

c) 100

d) 400

Ans (b )

Que 10

Conical vessel whose internal dimensions are 105 cm and 120 cm  in diameter is full of water if a cubic decimeter of water weight 1 kg 500 gram the weight of the water contained in the vessel is

a) 594

b ) 494

c) 444 kg

d) 394 kg

Ans ( a)

### Surface Area of Cube and Cuboid(ex 13.1)

Que 1

Plastic box 1.5 m long 1.25 m wide 65cm  deep is to be made it is opened at the top determine

Area of the sheet required for making the box

Find cost of sheet if the sheet measuring 1 metre square cost rupees 20

Solution--

Que 2

The length breadth and height of a room are 5 m 4 m and 3 m find the cost of whitewashing the walls of the room and the ceiling at the rate of 7.50 per metre square

Solution--

Que 3

The floor of a rectangular hall has a perimeter 250 If the cost of painting of  the four walls at the rate of 10 per m^2 is Rs 15000  find the height of the hall

Solution--

Basic concept of surface area of cube and cuboid

Cuboid

Length breadth and height all are not equal to each other

Surface Area = 2( LB+BH+LH)

Volume =L×B×H

Cube

in case of cube length breadth and height all are equal to each other

Surface Area = 6L*L

Volume = L*L*L

Watch  lecture in order to understand how to find surface area of cube and cuboid for any question

Cube
In this lecture
Various questions based on cube are discussed

Area of four walls

Formula

2H(L+B)
Watch the lecture

### Surface Area of Cylinder(ex 13.2)

Que 1

The curved surface area of right circular cylinder of height 14 cm is 88 cm ^2 find the diameter of the base of the cylinder

Ans 2 cm

Que 2

It is required to make a closed cylindrical tank of height 1 m and base diameter 140 cm from a metal sheet how many square metres of the sheet are required for the same

Ans 7.48m^2

Que 10

A lampshade is to be covered with a decorative cloth the frame has a base diameter of 20 cm and height of 30 cm margin of 2.5 cm is to be given for folding it over the top and bottom of the frame find how much cloth is required for covering the lamp shade

Que 11

The students of a vidyalaya where asked to participate in a competition for making and decorating welders in the shape of a cylinder with the base using cardboard each pen holder was to be of radius 3 cm and height 10.5 CM the weather was to supply the competitors with the cardboard if there are 35 competitors how much cardboard was required to be fought for the competition

Formula

Surface area of cylinder

2πrh

Part 2

Surface Area of Cone ( Ex 13.3)

Concept base Episode

Curved surface area of cone =1/3 πr*r *h

Exercise 13 .3

Que 1

Diameter of base of a cone is 10.5 cm and its slant height is 10 cm find its curved surface area

Que 2

Find the total surface area of a cone if its slant height is 21 m and diameter of its base is 24 m

Que 3

Curved surface area of a cone is 308 cm^2 and its slant height is 14 cm find the radius of the base and total surface area of the cone

Que 4

A conical tent is 10 m high radius of its base is 24 m find the slant height of the tent and the cost of Canvas required to make the tent if cost of 1 m is square Canvas is rupees 70

Que 5

What length of tarpaulin 3M wide will be required to make conical tent of height 8 m and base radius 6 m extra length of material that will be required in a stitching and wastage of the cloth is 20 cm is equal to 3.14

Que 6

The slant height and base diameter of a conical tomb are 25 m and 14 m respectively find the cost of whitewashing its curved surface area at the rate of rupees 210  per 100 m^2

Ex 13.4

Que 1
Find surface area of sphere of radius 10.5 cm

Que 2 find surface area of sphere of diameter 14 cm

Que 3. Find total surface area of hemisphere of radius 10 cm

Que 4
Radius of a spherical balloon increases from. 7 cm to 14 cm as air pushes in it find ratio of surface areas in  two cases

Que 5

A hemispherical bowl made of brasshas inner diameter 10.5 cm  . Find the cost of tin plating it on inside at the rate of Rs 16 per  100cm²

Que 6

Find the radius of a sphere whose surface area is 154 cm²

Solution of all above questions

Part 2

Que 7

The diameter of the moon is approximately one-fourth of the diameter of the earth. Find the ratio of their surface area

Que 8

A hemispherical bowl is made of steel 0.25 cm thick the inner radius of the bowl is
5 cm find the outer curved surface area of the bowl

Que 9

A right circular  cylinder just encloses a sphere of radius r .Find

1) Surface of sphere

2) Curved surface area of cylinder

3) Ratios of area of sphere and cylinder

Solution of above Problems

Ex 13.5

Que 1

A match box measure 4cm×2.5 cm × 1.5 cm what will be the volume of 12 such packets

Solution---

Que 2

A cuboidal water tank is 6m long 5 m wide and 4.5 m deep .How many litres of water can it hold ?

Solution

Que 3

A cuboidal vessel is 10 m long 8m wide. How high must it  to be made  to hold 380 cubic metres of liquid?

Solution

Que 4

Find cost of digging a cuboidal pit 8m long 6m broad ,3m deep at the rate of Rs 30 per m³

Solution

Que 5

The capacity of a cuboidal tank is 50000 litres of water  Find the breadth of the tank
If its length and depth are 2.5 m and 10 m

Solution

Que 6
A village having a population of 4000 requires 150 litres of water per head per day .It has a tank measuring 20m×15m×6m For how many days will the water  of this tank last?

Solution

Que 7

A godown measures 40m ×25m ×15 m .Find the maximum number of wooden crates each measuring 1.5m×1.25m ×0.5 m that can be stored in the go down ?

Solution

Que 8

A solid cube of side 12 cm is cut into 8 cubes of equal volume what will be the side of the new cube also find the ratio between their surface area

Solution--

Que 9

River 3 m Deep and 40m wide is flowing at the rate of 2 km per hour how much water will fall into the sea in a minute

Solution

Ex 13.6

Que 1

The circumference of a base of a cylindrical vessel is 132 cm and its height is 25 cm how many litres of water can it hold

Solution---

Que 2

The inner diameter of a cylindrical wooden pipe is 24 cm and its outer diameter is 28 cm the length of the pipe is 35 cm find the mass of the pipe if 1 cm³ of the wood has a mass of 0.6 g

Solution

Que 3

A soft drink is available in two packs

A rectangular can with rectangular base of length 5 cm and breadth 4cm having a height of 15 cm

Plastic cylinder with a circular base of diameter 7 cm and height 10cm

Which container has a greater capacity and by how much

Solution---

Que 4

If the lateral surface of a cylinder is  94.2 cm² and its height is 5 cm then find the radius of base and its volume

Solution

Que 5

It cost Rs 2200  to paint the inner curved surface of a cylindrical vessel 10 m deep  If  the cost of painting is at the rate of
Rs 25 per m²  Find
the inner curved surface area of vessel
capacity of the vessel

Solution---

Que 6

The capacity of a closed cylindrical vessel of height 1m is 15.4 litres how many square metres of a metal sheet would be needed to make it?

Solution---

Que 7

Lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior the diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm if the length of the pencil is 14 cm find the volume of the wood and that of the graphite

Solution---

Que 8

A patient in hospital is given soup daily in a cylindrical bowl of diameter 7 cm .If the bowl is filled with the soup to a height of 4 cm how much soup  the hospital has to prepare daily  to serve 250 patients

Solution --

Exercise 13.7

Que 1

Find the volume of right circular cone

1) radius = 6 cm ,height of 7 cm

2) radius 3.5 cm ,height 12 cm

Solution--

Que 2

Find the capacity in litres of a conical vessel

1) radius 7 cm ,slant height 25 cm

2) height 12 cm ,slant height 13 cm

Solution

Que 3

The height of a cone is 15 cm if its volume is 1570 cm³  find the radius of the base

Solution

Use formula

Volume = 1/3 πr²h

r= 10 cm

Que 4

If the volume of a right circular cone of 9cm is 48 π cm³ find the diameter of its base

Ans -- 8 cm

Solution

Que 6

Que 7 and 8

Que 9

A conical pit of top diameter 3.5 m is 12 m deep what is its capacity in kilo litres

Ans -- 38.5 kl

Solution

Que 10

Ex 13 .8

Que 1
Find the volume of a sphere whose radius is

a) 7 cm.            b) 0.63m

Solution---

Que 2

Find the amount of water displaced by a solid spherical ball of diameter

a) 28cm.         b) 0.21 m

Solution---

Que 3

Que 4

Que 5

Que 6

Que 7

## Heron's Formula

Heroine was born in about 10AD  in Alexandria in Egypt  about in applied mathematics yes derive the famous formula for the area of triangle in the terms of three sides this formula is given by heroines so it is also known as heron  formula this formula is very helpful in finding the area of triangle when it is not possible to find the height of the triangle

Concept in one shot

Que 1
The semi perimeter of a triangle having the length of its sides is 20 cm and 15 cm and 9 cm
a) 44cm
b) 21 cm
c) 22cm
d) none

Ans ( c)

Que 2
The angles of a triangle are in ratio  5:3:7 the triangle is
a) An acute angled triangle
b) An obtuse angled triangle
c) A right triangle
d) an isosceles triangle

Ans ( b)

Que 3
Length of perpendicular drawn on smallest side of a scalene triangle is
a) largest
b) smallest
c) no relation
d) equal

Ans ( a)

Que 4
Semi perimeter of a scalene triangle of  side  k,2k,3k
a) k
b) 2k
c) 3k
d) 4k

Ans (c)

Que 5
The perimeter of a equilateral triangle is
60 m the area is
a) 10√3
b) 15√3
c)100√3
d) 50√3

Ans (c)

Que 6
The area of an equilateral triangle with side 6cm
a) 216
b) 18
c)9√3
d) 36√3

Ans (c)

Que 7
Area of the equilateral triangle is 16√3cm^2 then side of equilateral triangle is
a) 8cm
b) 4 cm
c) 16 cm
d) 24 cm

Ans (a)

Que 8
The sides of a triangle are 35 cm 54 cm and 61 cm respectively the length of its longest altitude is
a) 19√5
b) 24√5
c) 28
d) 10√5

Ans (b)

Que 9
The area of an isosceles triangle having base 2 cm and the length of one of equal sides is 4 cm
a) √15
b) √15/2
c) 2√15
d) 15

Ans (a)

Que 10
An isosceles right triangle has area 8 cm square the length of its hypotenuse is
a) √32
b) √16
c) √48
d) √24

Ans ( a )

### Exercise 12.1

Questions Covered in Attached lecture are

Que -- A traffic signal board indicating "school ahead "is an equilateral triangle of side a  cm find the area of the signal board using heron's  formula if its perimeter is 180 cm what will be the area of the signal board km

Solution

Que -- Find the area of the triangle two sides of which are 18 cm 10 cm and the perimeter is 42 cm

Que --- Sides of a triangle are in ratio

12 :17:25 and its perimeter is 540 cm find its area

Que --- An isosceles triangle has perimeter 30 cm each of the equal sides is 12 cm find the area of triangle

Solution

Solutions of all questions can be watched by clicking the recorded lecture given below

Exercises 12.2

Question Explained in attached Lecture are

Que 1
A park in the shape of quadrilateral ABCD has  angle C= 90° ,AB = 9 m ,BC = 12 m CD= 5m , and AD = 5 cm  how much area does it occupy

Solution

Que 2
Find the area of a quadrilateral  ABCD in which AB= 3 cm ,BC= 4 cm ,CD = 4 cm ,DA= 5 cm and AC = 5 cm

Que 5
Rhombus shaped field has green grass for 18 cows to graze if each side of rhombus is 30 and its longer diagonal is 48 metre how much area of grass field will each cow be getting

Solution --

Que 6
Umbrella is made by stitching 10 triangular pieces of cloth of two different colours each piece measuring 20 CM 50 cm and 50 cm how much cloth of each colour is required for the umbrella

Que 7

A kite in the shape of a square with diagonal 32 cm and an isosceles triangle of base 8 cm and side 6 cm each is to be made of different shades how much paper of each shade has been used in it

Que 8

A floral design on a floor is made up of 16 tiles which are triangular the sides of the triangle mean lines and 28 cm and 35 cm find the cost of polishing the tiles at the rate of 50 Paisa per cm  square

Solution
Solution of above all three questions can be watched in the attached lecture

Que 4
10:46
A triangle and parallelogram have the same base and same area at the sides of the triangle are 26 and 28 cm and 13 cm and the parallelogram is stands  on the base 28 cm find the height of the parallelogram

Solution ---

Que 9
22:25 (Time stamp in video)

A field is  in the shape of trapezium whose parallel sides are 25m and 10m and non parallel sides are 14 and 13 m find the area of the field

Solution--

### Formula

Area of a triangle with sides a b c
=✓s(s-a)(s-b)(s-c)

Where s is semiperimeter

s= (a+b+c)/2

## Mock Test Series Maths - Class 9 - Triangles

### Que 1Exterior angle of a triangle is equal to the sum of the twoa) Exterior anglesb) Interior anglesc) Interior opposite anglesd) Alternate anglesAns (c)Que 2It is not possible to construct a triangle when its sides area) 8.3 ,3.4,6.1b) 5.4,2.3,3.1c) 6,7,10d) 3,5,5Ans ( b) Que 3In triangle ABC £B=30° ,£C= 80° £A= 70°a) AB>BC>AC b) AB <BC<ACc) AB<BC>ACd) AB >BC<ACAns (a)Que 4In a triangle ABC ,AB=AC, £A=80°  The sum of two remaining angles area) 100°b) 50°c) 130°d) 180 °Ans ( a)Que 5In triangle ABC ,£A= 50°,£B = 60 ° then greatest side is equal to a) ABb) BCc) ACd) cannot sayAns (a)Que 6 One of the angles of triangle is 75 degree if the difference of other two angle is 35 degree then the largest angle of triangle has a measure ofa) 80°b) 75°c) 100°d) 135°Ans (b)Que 7An exterior angle of a triangle is 120 degree and the two interior opposite angles are equal each of these angles isa)60°b) 100°c) 120°d) 180°Ans ( a )Que 8In triangle ABC if £B<£C thena) AB>ACb) AB=ACc) AB<ACd) none of aboveAns (a )Que 9 A triangle can havea) Two right anglesb) Two obtuse anglesc) All angles more than 60 °d ) Two acute anglesAns (d)Que 10In a right triangle the two acute angles are in ratio 4 : 5 The acute angles area) 50°,40°b)40°,50°c) 45°,55°d) 30 °,60°Ans (b)Solution of exercise of Ncert

Exercise 7.1

Que 1

Show that ∆ABC =~∆ ABD

What can you say about BC and BD

Solution -- watch the lecture

Que 2

ABCD is a quadrilateral in which AD = BC and  angle DAB = angle CBA prove that

1) ∆ABD =~ ∆ BAC

2) BD = AC

3) angle ABD = angle BAC

Solution

Watch the lecture

Que 3

AD and BC are equal perpendiculars to a line segment  AB show that  CD bisects AB

Solution---

Que 4 --- l and m are two parallel lines intersected by another pair of parallel lines p and q .Show that ∆ABC =~∆CDA

Solution--

Que 5

Line.  l  is the bisector of an angle £A  and B  is any point on  l and BP and BQ  are perpendiculars from to the arms of £A show that

1) ∆APB =~∆AQB

2) BP =BQ or B is equidistant from the arms of angle A

Solution--

Que 6

AC= AE ,AB =AD and angle BAD =angle EAC .Show that BC = DE

Solution----

Que 7

AB  is a line segment and p is its midpoint D and E  are the points on the same side of AB  such that  £BAD =£ABE and £EPA = £DPB  show that

1) ∆DAP =~∆EBP

Solution--

Que 8

In right triangle  ABC right angled at C,M   is the midpoint of hypotenuse  AB .C is joined to M and produced to a point D  such that DM = CM point D   is joined to  point B  show that

1) ∆AMC =~ ∆BMD

2) angle DBC is a right angle

3) ∆DBC =~∆ACB

4) CM = 1/2 AB

Solution----

Exercise 7.2

Questions discussed in attached lecture are

Que 3

ABC is an isosceles triangle in which altitudes BE  and CF are drawn to the equal sides AC and AB  respectively show that these attitudes are equal

Que 4

ABC is a triangle in which attitudes BE and CF  to the sides AC and AB  are equal show that

1) ∆ABE =~ ∆ ACF

2) AB = AC ie ABC is an isosceles triangle

Que 5

ABC and DBC are two  isosceles triangles on the same base  BC show that £ABD =£ACD

Solution--

Que 7
ABC is right angle triangle in which  angle  A is 90° and AB = AC find angle B and angle C

Solution ---

In equality concept base

Exercise 7.3

Que 1

∆ABC and ∆DBC are two isosceles triangles on the same base BC  and vertices A and D  are on the same side  of BC if AD  is extended to intersect BC  at P  show that

1) ∆ABD =~ ∆ ACD

2) ∆ABP =~∆ACP

3) AP bisects angle A as well as angle D

4) AP is the perpendicular bisector of BC

Solution ----

Exercise 7.4

Que 1

Show that in a right angle triangle the hypotenuse is the longest side

Que 2 -- Sides AB and AC of ∆ ABC are extended to the points Pand Q  respectively Also £PBC <£QCB .Show that AC> AB

Que 3
£B <£Aand £C<£D .Show that AD <BC

Summary

SAS Congruence Rule --
If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle then the two triangles are congruent

ASA Congruence Rule ---

Is to engage and included side of a triangle are equal to two angles and the included side of other triangle then the two triangles are congruent

SSS  Congruence Rule --

Three sides of one triangle are equal to three sides of the other triangle then the two triangles are congruent

RHS Congruence Rule --

If in two right triangles hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangle then the two triangles are congruent

In a triangle angle opposite to the longer side is larger

In a triangle side opposite to the larger angle is longer

Sum of any two sides of a triangle is greater than the third side

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