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Parallelograms and Triangles Class 9 Questions And Answers are discusses here. These NCERT solutions are developed by expert team at Careers360 team keeping in mind latest CBSE syllabus 2023-24. These solutions are simple, easy to understand and cover all the concepts step by step thus, ultimately help the students. In this particular NCERT book chapter, you will learn about the areas of different triangles and parallelograms.
NCERT solutions for class 9 maths chapter 9 Areas Of Parallelograms And Triangles is also covering the solutions to the application based questions as well. There are several interesting problems in the class 9 maths NCERT syllabus . Solving these problems will help you in improving the concepts of the chapter and also in exams like the Olympiads. In total there are 4 practice exercises having 51 questions. Areas of parallelograms and triangles class 9 NCERT solutions have covered all the exercises including the optional ones in a detailed manner. Here you will get NCERT solutions for class 9 Maths also.
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Areas of Parallelograms and Triangles Class 9 Questions And Answers PDF Free Download
Area of Parallelogram = Base × Height
Area of Triangle = (1/2) × Base × Height
Alternatively, it can be expressed as half the area of the parallelogram containing the triangle:
Area of Triangle = (1/2) × Area of Parallelogram
Area of Trapezium = (1/2) × (Sum of Parallel Sides) × Distance between Parallel Sides
Area of Rhombus = (1/2) × (Product of Diagonals)
Free download NCERT Solutions for Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles for CBSE Exam.
Class 9 maths chapter 9 question answer - Exercise: 9.1
Answer:
In figure (i), (iii) and (v) we can see that. they lie on the same base and between the same parallel lines.
In figure (i) figure (iii) figure (v)
Common base DC QR AD
Two parallels DC and AB QR and PS AD and BQ
Class 9 areas of parallelograms and triangles NCERT solutions - Exercise : 9.2
Q1 In Fig. , ABCD is a parallelogram, and . If , and , find AD.
Answer:
We have,
AE DC and CF AD
AB = 16 cm, AE = 8 cm and CF = 10 cm
Since ABCD is a parallelogram,
therefore, AB = DC = 16 cm
We know that, area of parallelogram (ABCD) = base . height
= CD AE = (16 8 )
SInce, CF AD
therefore area of parallelogram = AD CF = 128
= AD = 128/10
= AD = 12.8 cm
Thus the required length of AD is 12.8 cm.
Q2 If E, F, G and H are respectively the mid-points of the sides of a parallelogram ABCD, show that
Answer:
Join GE and HE,
Since F, F, G, H are the mid-points of the parallelogram ABCD. Therefore, GE || BC ||AD and HF || DC || AB.
It is known that if a triangle and the parallelogram are on the same base and between the same parallel lines. then the area of the triangle is equal to the half of the area of the parallelogram.
Now, EFG and ||gm BEGC are on the same base and between the same parallels EG and BC.
Therefore, ar ( EFG) = ar (||gm BEGC)...............(i)
Similarly, ar ( EHG) = 1/2 . ar(||gm AEGD)..................(ii)
By adding eq (i) and eq (ii), we get
ar ( EFG) + ar ( EHG) = 1/2 (ar (||gm BEGC) + ar(||gm AEGD))
ar (EFGH) = 1/2 ar(ABCD)
Hence proved
Answer:
We have,
ABCD is a parallelogram, therefore AB || CD and BC || AD.
Now, APB and ||gm ABCD are on the same base AB and between two parallels AB and DC.
Therefore, ar ( APB) = 1/2 . ar(||gm ABCD)...........(i)
Also, BQC and ||gm ABCD are on the same base BC and between two parallels BC and AD.
Therefore, ar( BQC) = 1/2 . ar(||gmABCD)...........(ii)
From eq(i) and eq (ii), we get,
ar ( APB) = ar( BQC)
Hence proved.
Q4 (i) In Fig. , P is a point in the interior of a parallelogram ABCD. Show that
[ Hint : Through P, draw a line parallel to AB.]
Answer:
We have a ||gm ABCD and AB || CD, AD || BC. Through P, draw a line parallel to AB
Now, APB and ||gm ABEFare on the same base AB and between the same parallels EF and AB.
Therefore, ar ( APB) = 1/2 . ar(ABEF)...............(i)
Similarly, ar ( PCD ) = 1/2 . ar (EFDC) ..............(ii)
Now, by adding both equations, we get
Hence proved.
Q4 (ii) In Fig. , P is a point in the interior of a parallelogram ABCD. Show that
[ Hint: Through P, draw a line parallel to AB.]
Answer:
We have a ||gm ABCD and AB || CD, AD || BC. Through P, draw a line parallel to AB
Now, APD and ||gm ADGHare on the same base AD and between the same parallels GH and AD.
Therefore, ar ( APD) = 1/2 . ar(||gm ADGH).............(i)
Similarily, ar ( PBC) = 1/2 . ar(||gm BCGH)............(ii)
By adding the equation i and eq (ii), we get
Hence proved.
Q5 In Fig. 9.17, PQRS and ABRS are parallelograms and X is any point on side BR. Show that
(i)
(ii)
Answer:
(i) Parallelogram PQRS and ABRS are on the same base RS and between the same parallels RS and PB.
Therefore, ............(i)
Hence proved
(ii) AXS and ||gm ABRS are on the same base AS and between same parallels AS and RB.
Therefore, ar ( AXS) = 1/2 . ar(||gm ABRS)............(ii)
Now, from equation (i) and equation (ii), we get
Hence proved.
Answer:
We have a field in the form of parallelogram PQRS and a point A is on the side RS. Join AP and AQ. The field is divided into three parts i.e, APS, QAR and PAQ.
Since APQ and parallelogram, PQRS is on the same base PQ and between same parallels RS and PQ.
Therefore, ............(i)
We can write above equation as,
ar (||gm PQRS) - [ar ( APS) + ar( QAR)] = 1/2 .ar(PQRS)
from equation (i),
Hence, she can sow wheat in APQ and pulses in [ APS + QAR] or wheat in [ APS + QAR] and pulses in APQ.
Class 9 maths chapter 9 NCERT solutions - exercise : 9.3
Q1 In Fig. , E is any point on median AD of a . Show that.
Answer:
We have ABC such that AD is a median. And we know that median divides the triangle into two triangles of equal areas.
Therefore, ar( ABD) = ar( ACD)............(i)
Similarly, In triangle BEC,
ar( BED) = ar ( DEC)................(ii)
On subtracting eq(ii) from eq(i), we get
ar( ABD) - ar( BED) =
Hence proved.
Q2 In a triangle ABC, E is the mid-point of median AD. Show that .
Answer:
We have a triangle ABC and AD is a median. Join B and E.
Since the median divides the triangle into two triangles of equal area.
ar( ABD) = ar ( ACD) = 1/2 ar( ABC)..............(i)
Now, in triangle ABD,
BE is the median [since E is the midpoint of AD]
ar ( BED) = 1/2 ar( ABD)........(ii)
From eq (i) and eq (ii), we get
ar ( BED) = 1/2 . (1/2 ar(ar ( ABC))
ar ( BED) = 1/4 .ar( ABC)
Hence proved.
Q3 Show that the diagonals of a parallelogram divide it into four triangles of equal area.
Answer:
Let ABCD is a parallelogram. So, AB || CD and AD || BC and we know that Diagonals bisects each other. Therefore, AO = OC and BO = OD
Since OD = BO
Therefore, ar ( BOC) = ar ( DOC)...........(a) ( since OC is the median of triangle CBD)
Similarly, ar( AOD) = ar( DOC) ............(b) ( since OD is the median of triangle ACD)
and, ar ( AOB) = ar( BOC)..............(c) ( since OB is the median of triangle ABC)
From eq (a), (b) and eq (c), we get
ar ( BOC) = ar ( DOC)= ar( AOD) = ( AOB)
Thus, the diagonals of ||gm divide it into four equal triangles of equal area.
Answer:
We have, ABC and ABD on the same base AB. CD is bisected by AB at point O.
OC = OD
Now, in ACD, AO is median
ar ( AOC) = ar ( AOD)..........(i)
Similarly, in BCD, BO is the median
ar ( BOC) = ar ( BOD)............(ii)
Adding equation (i) and eq (ii), we get
ar ( AOC) + ar ( BOC) = ar ( AOD) + ar ( BOD)
Hence proved.
Answer:
We have a triangle ABC such that D, E and F are the midpoints of the sides BC, CA and AB respectively.
Now, in ABC,
F and E are the midpoints of the side AB and AC.
Therefore according to mid-point theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half of the third side.
EF || BC or EF || BD
also, EF = 1/2 (BC)
[ D is the midpoint of BC]
Similarly, ED || BF and ED = FB
Hence BDEF is a parallelogram.
Q5 (ii) D, E and F are respectively the mid-points of the sides BC, CA and AB of a . Show that
Answer:
We already proved that BDEF is a ||gm.
Similarly, DCEF and DEAF are also parallelograms.
Now, ||gm BDEF and ||gm DCEF is on the same base EF and between same parallels BC and EF
Ar (BDEF) = Ar (DCEF)
Ar( BDF) = Ar ( DEF) .............(i)
It is known that diagonals of ||gm divides it into two triangles of equal area.
Ar(DCE) = Ar (DEF).......(ii)
and, Ar( AEF) = Ar ( DEF)...........(iii)
From equation(i), (ii) and (iii), we get
Ar( BDF) = Ar(DCE) = Ar( AEF) = Ar ( DEF)
Thus, Ar ( ABC) = Ar( BDF) + Ar(DCE) + Ar( AEF) + Ar ( DEF)
Ar ( ABC) = 4 . Ar( DEF)
Hence proved.
Q5 (iii) D, E and F are respectively the mid-points of the sides BC, CA and AB of a . Show that
Answer:
Since we already proved that,
ar( DEF) = ar ( BDF).........(i)
So, ar(||gm BDEF) = ar( BDF) + ar ( DEF)
= 2 . ar( DEF) [from equation (i)]
Hence proved.
Q6 (i) In Fig. , diagonals AC and BD of quadrilateral ABCD intersect at O such that. If , then show that:
[ Hint: From D and B, draw perpendiculars to AC.]
Answer:
We have ABCD is quadrilateral whose diagonals AC and BD intersect at O. And OB = OD, AB = CD
Draw DE AC and FB AC
In DEO and BFO
DOE = BOF [vertically opposite angle]
OED = BFO [each ]
OB = OD [given]
Therefore, by AAS congruency
DEO BFO
DE = FB [by CPCT]
and ar( DEO) = ar( BFO) ............(i)
Now, In DEC and ABF
DEC = BFA [ each ]
DE = FB
DC = BA [given]
So, by RHS congruency
DEC BFA
1 = 2 [by CPCT]
and, ar( DEC) = ar( BFA).....(ii)
By adding equation(i) and (ii), we get
Hence proved.
Q6 (ii) In Fig. , diagonals AC and BD of quadrilateral ABCD intersect at O such that . If , then show that:
[ Hint: From D and B, draw perpendiculars to AC.]
Answer:
We already proved that,
Now, add on both sides we get
Hence proved.
Q6 (iii) In Fig. , diagonals AC and BD of quadrilateral ABCD intersect at O such that . If , then show that:
or ABCD is a parallelogram.
[ Hint : From D and B, draw perpendiculars to AC.]
Answer:
Since DCB and ACB both lie on the same base BC and having equal areas.
Therefore, They lie between the same parallels BC and AD
CB || AD
also 1 = 2 [ already proved]
So, AB || CD
Hence ABCD is a || gm
Q7 D and E are points on sides AB and AC respectively of such that . Prove that .
Answer:
We have ABC and points D and E are on the sides AB and AC such that ar( DBC ) = ar ( EBC)
Since DBC and EBC are on the same base BC and having the same area.
They must lie between the same parallels DE and BC
Hence DE || BC.
Answer:
We have a ABC such that BE || AC and CF || AB
Since XY || BC and BE || CY
Therefore, BCYE is a ||gm
Now, The ||gm BCEY and ABE are on the same base BE and between the same parallels AC and BE.
ar( AEB) = 1/2 .ar(||gm BEYC)..........(i)
Similarly, ar( ACF) = 1/2 . ar(||gm BCFX)..................(ii)
Also, ||gm BEYC and ||gmBCFX are on the same base BC and between the same parallels BC and EF.
ar (BEYC) = ar (BCFX).........(iii)
From eq (i), (ii) and (iii), we get
ar( ABE) = ar( ACF)
Hence proved.
Answer:
Join the AC and PQ.
It is given that ABCD is a ||gm and AC is a diagonal of ||gm
Therefore, ar( ABC) = ar( ADC) = 1/2 ar(||gm ABCD).............(i)
Also, ar( PQR) = ar( BPQ) = 1/2 ar(||gm PBQR).............(ii)
Since AQC and APQ are on the same base AQ and between same parallels AQ and CP.
ar( AQC) = ar ( APQ)
Now, subtracting ABQ from both sides we get,
ar( AQC) - ar ( ABQ) = ar ( APQ) - ar ( ABQ)
ar( ABC) = ar ( BPQ)............(iii)
From eq(i), (ii) and (iii) we get
Hence proved.
Q10 Diagonals AC and BD of a trapezium ABCD with intersect each other at O. Prove that .
Answer:
We have a trapezium ABCD such that AB || CD and it's diagonals AC and BD intersect each other at O
ABD and ABC are on the same base AB and between same parallels AB and CD
ar( ABD) = ar ( ABC)
Now, subtracting AOB from both sides we get
ar ( AOD) = ar ( BOC)
Hence proved.
Q11 In Fig. , ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show that
(i)
(ii)
Answer:
We have a pentagon ABCDE in which BF || AC and CD is produced to F.
(i) Since ACB and ACF are on the same base AC and between same parallels AC and FB.
ar( ACB) = ar ( ACF)..................(i)
(ii) Adding the ar (AEDC) on both sides in equation (i), we get
ar( ACB) + ar(AEDC) = ar ( ACF) + ar(AEDC)
Hence proved.
Answer:
We have a quadrilateral shaped plot ABCD. Draw DF || AC and AF || CF.
Now, DAF and DCF are on the same base DF and between same parallels AC and DF.
ar ( DAF) = ar( DCF)
On subtracting DEF from both sides, we get
ar( ADE) = ar( CEF)...............(i)
The portion of ADE can be taken by the gram panchayat and on adding the land CEF to his (Itwaari) land so as to form a triangular plot.( ABF)
We need to prove that ar( ABF) = ar (quad. ABCD)
Now, adding ar(quad. ABCE) on both sides in eq (i), we get
ar ( ADE) + ar(quad. ABCE) = ar( CEF) + ar(quad. ABCE)
ar (ABCD) = ar( ABF)
Answer:
We have a trapezium ABCD, AB || CD
XY ||AC meets AB at X and BC at Y. Join XC
Since ADX and ACX lie on the same base CD and between same parallels AX and CD
Therefore, ar( ADX) = ar( ACX)..........(i)
Similarly ar( ACX) = ar( ACY).............(ii) [common base AC and AC || XY]
From eq (i) and eq (ii), we get
ar( ADX) = ar ( ACY)
Hence proved.
Answer:
We have, AP || BQ || CR
BCQ and BQR lie on the same base (BQ) and between same parallels (BQ and CR)
Therefore, ar ( BCQ) = ar ( BQR)........(i)
Similarly, ar ( ABQ) = ar ( PBQ) [common base BQ and BQ || AP]............(ii)
Add the eq(i) and (ii), we get
ar ( AQC) = ar ( PBR)
Hence proved.
Answer:
We have,
ABCD is a quadrilateral and diagonals AC and BD intersect at O such that ar( AOD) = ar ( BOC) ...........(i)
Now, add ar ( BOA) on both sides, we get
ar( AOD) + ar ( BOA) = ar ( BOA) + ar ( BOC)
ar ( ABD) = ar ( ABC)
Since the ABC and ABD lie on the same base AB and have an equal area.
Therefore, AB || CD
Hence ABCD is a trapezium.
Q16 In Fig. , and . Show that both the quadrilaterals ABCD and DCPR are trapeziums.
Answer:
Given,
ar( DPC) = ar( DRC) ..........(i)
and ar( BDP) = ar(ARC)............(ii)
from equation (i),
Since DRC and DPC lie on the same base DC and between same parallels.
CD || RP (opposites sides are parallel)
Hence quadrilateral DCPR is a trapezium
Now, by subtracting eq(ii) - eq(i) we get
ar( BDP) - ar( DPC) = ar( ARC) - ar( DRC)
ar( BDC) = ar( ADC) (Since theya are on the same base DC)
AB || DC
Hence ABCD is a trapezium.
Areas of parallelograms and triangles class 9 solutions - Exercise : 9.4
Answer:
We have ||gm ABCD and a rectangle ABEF both lie on the same base AB such that, ar(||gm ABCD) = ar(ABEF)
for rectangle, AB = EF
and for ||gm AB = CD
CD = EF
AB + CD = AB + EF ...........(i)
SInce BEC and AFD are right angled triangle
Therefore, AD > AF and BC > BE
(BC + AD ) > (AF + BE)...........(ii)
Adding equation (i) and (ii), we get
(AB + CD)+(BC + AD) > (AB + BE) + (AF + BE)
(AB + BC + CD + DA) > (AB + BE + EF + FA)
Hence proved, perimeter of ||gm ABCD is greater than perimeter of rectangle ABEF.
Q2 In Fig. , D and E are two points on BC such that . Show that .
Answer:
In ABC, D and E are two points on BC such that BD = DE = EC
AD is the median of ABE, therefore,
area of ABD= area of AED...................(1)
AE is the median of ACD, therefore,
area of AEC= area of AED...................(2)
From (1) and (2)
area of ABD=area of AED= area of AEC
Hence proved.
Q3 In Fig. , ABCD, DCFE and ABFE are parallelograms. Show that .
Answer:
Given,
ABCD is a parallelogram. Therefore, AB = CD & AD = BC & AB || CD .......(i)
Now, ADE and BCF are on the same base AD = BC and between same parallels AB and EF.
Therefore, ar ( ADE) = ar( BCF)
Hence proved.
Answer:
Given,
ABCD is a ||gm and AD = CQ. Join AC.
Since DQC and ACD lie on the same base QC and between same parallels AD and QC.
Therefore, ar( DQC) = ar( ACD).......(i)
Subtracting ar( PQC) from both sides in eq (i), we get
ar( DPQ) = ar( PAC).............(i)
Since PAC and PBC are on the same base PC and between same parallel PC and AB.
Therefore, ar( PAC) = ar( PBC)..............(iii)
From equation (ii) and eq (ii), we get
Hence proved.
[ Hint: Join EC and AD. Show that and , etc.]
Answer:
Let join the CE and AD and draw . It is given that ABC and BDE is an equilateral triangle.
So, AB =BC = CA = and D i sthe midpoint of BC
therefore,
(i) Area of ABC = and
Area of BDE =
Hence,
[ Hint: Join EC and AD. Show that and , etc.]
Answer:
Since ABC and BDE are equilateral triangles.
Therefore, ACB = DBE =
BE || AC
BAE and BEC are on the same base BE and between same parallels BE and AC.
Therefore, ar ( BAE) = ar( BEC)
ar( BAE) = 2 ar( BED) [since D is the meian of BEC ]
Hence proved.
[ Hint : Join EC and AD. Show that and , etc.]
Answer:
We already proved that,
ar( ABC) = 4.ar( BDE) (in part 1)
and, ar( BEC) = 2. ar( BDE) (in part ii )
ar( ABC) = 2. ar( BEC)
Hence proved.
Answer:
Since ABC and BDE are equilateral triangles.
Therefore, ACB = DBE =
BE || AC
BDE and AED are on the same base ED and between same parallels AB and DE.
Therefore, ar( BED) = ar( AED)
On subtracting EFD from both sides we get
Hence proved.
[ Hint : Join EC and AD. Show that and , etc.]
Answer:
In right angled triangle ABD, we get
So, in PED,
So,
Therefore, the Area of ..........(i)
And, Area of triangle ...........(ii)
From eq (i) and eq (ii), we get
ar( AFD) = 2. ar( EFD)
Since ar( AFD) = ar( BEF)
[ Hint: Join EC and AD. Show that and , etc.]
Answer:
.....(from part (v) ar( BFE) = 2. ar( FED) ]
Hence proved.
Q6 Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that
[ Hint: From A and C, draw perpendiculars to BD.]
Answer:
Given that,
A quadrilateral ABCD such that it's diagonal AC and BD intersect at P. Draw and
Now, ar( APB) = and,
Therefore, ar( APB) ar( CDP)=
....................(i)
Similarly, ar( APD) ar ( BPC) =
................(ii)
From eq (i) and eq (ii), we get
Hence proved.
Answer:
We have ABC such that P, Q and R are the midpoints of the side AB, BC and AP respectively. Join PQ, QR, AQ, PC and RC as shown in the figure.
Now, in APC,
Since R is the midpoint. So, RC is the median of the APC
Therefore, ar( ARC) = 1/2 . ar ( APC)............(i)
Also, in ABC, P is the midpoint. Thus CP is the median.
Therefore, ar( APC) = 1/2. ar ( ABC)............(ii)
Also, AQ is the median of ABC
Therefore, 1/2. ar ( ABC) = ar (ABQ)............(iii)
In APQ, RQ is the median.
Therefore, ar ( PRQ) = 1/2 .ar ( APQ).............(iv)
In ABQ, PQ is the median
Therefore, ar( APQ) = 1/2. ar( ABQ).........(v)
From eq (i),
...........(vi)
Now, put the value of ar( APC) from eq (ii), we get
Taking RHS;
(from equation (iii))
(from equation (v))
(from equation (iv))
Hence proved.
Answer:
In RBC, RQ is the median
Therefore ar( RQC) = ar( RBQ)
= ar (PRQ) + ar (BPQ)
= 1/8 (ar ABC) + ar( BPQ) [from eq (vi) & eq (A) in part (i)]
= 1/8 (ar ABC) + 1/2 (ar PBC) [ since PQ is the median of BPC]
= 1/8 (ar ABC) + (1/2).(1/2)(ar ABC) [CP is the medain of ABC]
= 3/8 (ar ABC)
Hence proved.
Answer:
QP is the median of ABQ
Therefore, ar( PBQ) = 1/2. (ar ABQ)
= (1/2). (1/2) (ar ABC) [since AQ is the median of ABC
= 1/4 (ar ABC)
= ar ( ARC) [from eq (A) of part (i)]
Hence proved.
Answer:
We have, a ABC such that BCED, ACFG and ABMN are squares on its side BC, CA and AB respec. Line segment meets BC at Y
(i) [each 90]
Adding on both sides, we get
In ABD and MBC, we have
AB = MB
BD = BC
Therefore, By SAS congruency
ABD MBC
Hence proved.
Answer:
SInce ||gm BYXD and ABD are on the same base BD and between same parallels BD and AX
Therefore, ar( ABD) = 1/2. ar(||gm BYXD)..........(i)
But, ABD MBC (proved in 1st part)
Since congruent triangles have equal areas.
Therefore, By using equation (i) we get
Hence proved.
Answer:
Since, ar (||gm BYXD) = 2 .ar ( MBC) ..........(i) [already proved in 2nd part]
and, ar (sq. ABMN) = 2. ar ( MBC)............(ii)
[Since ABMN and AMBC are on the same base MB and between same parallels MB and NC]
From eq(i) and eq (ii), we get
Answer:
[both 90]
By adding ABC on both sides we get
ABC + FCA = ABC + BCE
FCB = ACE
In FCB and ACE
FC = AC [sides of square]
BC = AC [sides of square]
FCB = ACE
FCB ACE
Hence proved.
Answer:
Since ||gm CYXE and ACE lie on the same base CE and between the same parallels CE and AX.
Therefore, 2. ar( ACE) = ar (||gm CYXE)
B ut, FCB ACE (in iv part)
Since the congruent triangle has equal areas. So,
Hence proved.
Answer:
Since ar( ||gm CYXE) = 2. ar( ACE) {in part (v)}...................(i)
Also, FCB and quadrilateral ACFG lie on the same base FC and between the same parallels FC and BG.
Therefore, ar (quad. ACFG) = 2.ar( FCB ) ................(ii)
From eq (i) and eq (ii), we get
Hence proved.
Answer:
We have,
ar(quad. BCDE) = ar(quad, CYXE) + ar(quad. BYXD)
= ar(quad, CYXE) + ar (quad. ABMN) [already proved in part (iii)]
Thus,
Hence proved.
Maths chapter 9 class 9 - Important Points
Interested students can practice class 9 maths ch 9 question answer using the following links
Chapter No. | Chapter Name |
Chapter 1 | Number Systems |
Chapter 2 | Polynomials |
Chapter 3 | Coordinate Geometry |
Chapter 4 | Linear Equations In Two Variables |
Chapter 5 | Introduction to Euclid's Geometry |
Chapter 6 | Lines And Angles |
Chapter 7 | Triangles |
Chapter 8 | Quadrilaterals |
Chapter 9 | Areas of Parallelograms and Triangles |
Chapter 10 | Circles |
Chapter 11 | Constructions |
Chapter 12 | Heron’s Formula |
Chapter 13 | Surface Area and Volumes |
Chapter 14 | Statistics |
Chapter 15 | Probability |
How To Use NCERT Solutions For Class 9 Maths Chapter 9 Areas Of Parallelograms And Triangles
Keep working hard and happy learning!
Areas of Parallelograms and Triangles, figures with the same base and between parallel lines, are the important topics of this chapter. Students can practice NCERT solutions for class 9 maths to command these concepts which are very important for the exam.
NCERT solutions are not only helpful for the students if they stuck while solving NCERT problems but also, these solutions are provided in a very detailed manner which will give them conceptual clarity.
Here you will get the detailed NCERT solutions for class 9 maths by clicking on the link. you can practice these ncert solutions for class 9 maths chapter 9 to command the concepts which give the confidence during exams.
In order to be able to tackle various types of questions that appear in exams, it is crucial to study and practice all the questions included in the NCERT Solutions for Class 9 Maths Chapter 9. The solutions are presented in a student-friendly language that facilitates easier comprehension of complex problems. These solutions are designed by Careers360 experts who possess extensive knowledge of mathematics.
Exam Date:28 April,2024 - 28 April,2024
Exam Date:28 April,2024 - 28 April,2024
Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.
The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.
A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.
GIS officer work on various GIS software to conduct a study and gather spatial and non-spatial information. GIS experts update the GIS data and maintain it. The databases include aerial or satellite imagery, latitudinal and longitudinal coordinates, and manually digitized images of maps. In a career as GIS expert, one is responsible for creating online and mobile maps.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.
If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy, IIT Kharagpur, JMI New Delhi.
Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive.
Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.
Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.
Individuals who opt for a career as a stock analyst examine the company's investments makes decisions and keep track of financial securities. The nature of such investments will differ from one business to the next. Individuals in the stock analyst career use data mining to forecast a company's profits and revenues, advise clients on whether to buy or sell, participate in seminars, and discussing financial matters with executives and evaluate annual reports.
A Researcher is a professional who is responsible for collecting data and information by reviewing the literature and conducting experiments and surveys. He or she uses various methodological processes to provide accurate data and information that is utilised by academicians and other industry professionals. Here, we will discuss what is a researcher, the researcher's salary, types of researchers.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.
Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems.
A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.
A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
Highway Engineer Job Description: A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.
Are you searching for a Field Surveyor Job Description? A Field Surveyor is a professional responsible for conducting field surveys for various places or geographical conditions. He or she collects the required data and information as per the instructions given by senior officials.
Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.
A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.
Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.
The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.
An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.
Are you searching for an ‘Anatomist job description’? An Anatomist is a research professional who applies the laws of biological science to determine the ability of bodies of various living organisms including animals and humans to regenerate the damaged or destroyed organs. If you want to know what does an anatomist do, then read the entire article, where we will answer all your questions.
For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.
Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.
Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.
Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.
A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.
Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.
An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story.
They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.
Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.
Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.
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