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NCERT Solutions for Exercise 9.4 Class 9 Maths Chapter 9 - Areas Of Parallelograms And Triangles

NCERT Solutions for Exercise 9.4 Class 9 Maths Chapter 9 - Areas Of Parallelograms And Triangles

Edited By Safeer PP | Updated on Jul 29, 2022 06:22 PM IST

The NCERT Solutions for Class 9 Maths exercise 9.4 is an optional (not from the examination point of view) exercise which comprises of theorems related to the previous exercises which include concepts of parallelograms and triangle on the same base and between the same parallels, having questions related to areas of figure which includes the medians for triangles.

Few important concepts related to NCERT syllabus Class 9 Maths chapter 9 exercise 9.4 in order to solve this exercise are:

  • Parallelograms having the same base and are in between the same parallels have the same area, as because the height (distance between the two parallel lines) continues as before for the two parallelograms and they have same bases subsequently their regions are equivalent.

  • Triangle that has a same base or the equivalent base and are between same pair of parallel lines have equivalent area. This is on the grounds that we realize that the area of the triangle is a half of the result of base and height. Since the height (distance between the parallels) continues as before for the two triangles and they have same bases subsequently their area are equivalent.

Major questions in this exercise use a blend of at least two or more concepts.

Along with NCERT book Class 9 Maths chapter 9 exercise 9.4 the following exercises are also present.

Areas Of Parallelograms And Triangles Class 9 Chapter 9 Exercise: 9.1

Q1 Parallelogram ABCD and rectangle ABEF are on the same base AB and have equal areas. Show that the perimeter of the parallelogram is greater than that of the rectangle.

Answer:

15958704265621595870423378

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
\Rightarrow CD = EF
\Rightarrow AB + CD = AB + EF ...........(i)

SInce \Delta BEC and \Delta AFD are right angled triangle
Therefore, AD > AF and BC > BE
\Rightarrow (BC + AD ) > (AF + BE)...........(ii)

Adding equation (i) and (ii), we get

(AB + CD)+(BC + AD) > (AB + BE) + (AF + BE)
\Rightarrow (AB + BC + CD + DA) > (AB + BE + EF + FA)
Hence proved, perimeter of ||gm ABCD is greater than perimeter of rectangle ABEF.

Q2 In Fig. \small 9.30 , D and E are two points on BC such that \small BD=DE=EC . Show that \small ar(ABD)=ar(ADE)=ar(AEC) .

1640236583768Answer:

In \Delta 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. \small 9.31 , ABCD, DCFE and ABFE are parallelograms. Show that \small ar(ADE)=ar(BCF) .

1640236638372 Answer:

Given,
15958705591711595870555496
ABCD is a parallelogram. Therefore, AB = CD & AD = BC & AB || CD .......(i)

Now, \Delta ADE and \Delta BCF are on the same base AD = BC and between same parallels AB and EF.
Therefore, ar ( \Delta ADE) = ar( \Delta BCF)

Hence proved.

Q4 In Fig. \small 9.32 , ABCD is a parallelogram and BC is produced to a point Q such that \small AD=CQ . If AQ intersects DC at P, show that \small ar (BPC) = ar (DPQ) . [ Hint : Join AC.]

1640236665039 Answer:

15958706193791595870615704
Given,
ABCD is a ||gm and AD = CQ. Join AC.
Since \Delta DQC and \Delta ACD lie on the same base QC and between same parallels AD and QC.
Therefore, ar( \Delta DQC) = ar( \Delta ACD).......(i)

Subtracting ar( \Delta PQC) from both sides in eq (i), we get
ar( \Delta DPQ) = ar( \Delta PAC).............(i)

Since \Delta PAC and \Delta PBC are on the same base PC and between same parallel PC and AB.
Therefore, ar( \Delta PAC) = ar( \Delta PBC)..............(iii)

From equation (ii) and eq (ii), we get

\small ar (BPC) = ar (DPQ)

Hence proved.

Q5 (i) In Fig. \small 9.33 , ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F, show that

\small ar(BDE)=\frac{1}{4}ar(ABC)

[ Hint: Join EC and AD. Show that \small BE\parallel AC and \small DE\parallel AB , etc.]

1640236707923 Answer:

15958706735671595870671414
Let join the CE and AD and draw EP \perp BC . It is given that \Delta ABC and \Delta BDE is an equilateral triangle.
So, AB =BC = CA = a and D i sthe midpoint of BC
therefore, BD = \frac{a}{2}= DE = BE
(i) Area of \Delta ABC = \frac{\sqrt{3}}{4}a^2 and
Area of \Delta BDE = \frac{\sqrt{3}}{4}(\frac{a}{2})^2= \frac{\sqrt{3}}{16}a^2

Hence, \small ar(BDE)=\frac{1}{4}ar(ABC)


Q5 (ii) In Fig. \small 9.33 , ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F, show that
\small ar(BDE)=\frac{1}{2}ar(BAE)

[ Hint: Join EC and AD. Show that \small BE\parallel AC and \small DE\parallel AB , etc.]


1640236728299

Answer:

15958707000691595870698541
Since \Delta ABC and \Delta BDE are equilateral triangles.
Therefore, \angle ACB = \angle DBE = 60^0
\Rightarrow BE || AC

\Delta BAE and \Delta BEC are on the same base BE and between same parallels BE and AC.
Therefore, ar ( \Delta BAE) = ar( \Delta BEC)
\Rightarrow ar( \Delta BAE) = 2 ar( \Delta BED) [since D is the meian of \Delta BEC ]
\Rightarrow \small ar(BDE)=\frac{1}{2}ar(BAE)
Hence proved.

Q5 (iii) In Fig. \small 9.33 , ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F, show that
\small ar(ABC)=2ar(BEC)

[ Hint : Join EC and AD. Show that \small BE\parallel AC and \small DE\parallel AB , etc.]


1640236744718

Answer:

We already proved that,
ar( \Delta ABC) = 4.ar( \Delta BDE) (in part 1)
and, ar( \Delta BEC) = 2. ar( \Delta BDE) (in part ii )

\Rightarrow ar( \Delta ABC) = 2. ar( \Delta BEC)

Hence proved.

Q5 (iv) In Fig.9.33, ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F, show that
\small ar(BFE)=ar(AFD)
[ Hint: Join EC and AD. Show that \small BE\parallel AC and \small DE\parallel AB , etc.]


1640236764354

Answer:

15958707074701595870706529
Since \Delta ABC and \Delta BDE are equilateral triangles.
Therefore, \angle ACB = \angle DBE = 60^0
\Rightarrow BE || AC

\Delta BDE and \Delta AED are on the same base ED and between same parallels AB and DE.
Therefore, ar( \Delta BED) = ar( \Delta AED)

On subtracting \Delta EFD from both sides we get

\small ar(BFE)=ar(AFD)

Hence proved.

Q5 (v) In Fig. \small 9.33 , ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F, show that
\small ar(BFE)=2ar(FED)

[ Hint : Join EC and AD. Show that \small BE\parallel AC and \small DE\parallel AB , etc.]


1640236770366

Answer:

15958707161571595870714602
In right angled triangle \Delta ABD, we get

\\AB^2 = AD^2+BD^2\\ AD^2 = AB^2-BD^2
=a^2 - \frac{a^2}{4}=\frac{3a^2}{4}
AD=\frac{\sqrt{3}a}{2}

So, in \Delta PED,
PE^2 = DE^2-DP^2
\\=(\frac{a}{2})^2-(\frac{a}{4})^2\\ =\frac{3a^2}{16}
So, PE=\frac{\sqrt{3}a}{4}

Therefore, the Area of \Delta AFD =1/2 (FD)\frac{\sqrt{3}a}{2} ..........(i)

And, Area of triangle \Delta EFD =1/2 (FD)\frac{\sqrt{3}a}{4} ...........(ii)

From eq (i) and eq (ii), we get
ar( \Delta AFD) = 2. ar( \Delta EFD)
Since ar( \Delta AFD) = ar( \Delta BEF)

\Rightarrow \small ar(BFE)=2ar(FED)


Q5 (vi) In Fig. 9.33 , ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F, show that
ar (FED) = \frac{1}{8} ar (AFC)

[ Hint: Join EC and AD. Show that BE\parallel AC and DE\parallel AB , etc.]

1640236806124 Answer:

ar(\Delta AFC) =
\\= ar(\Delta AFD) + ar(\Delta ADC)\\ = ar(\Delta BFE) + 1/2. ar(\Delta ABC)\\ = ar(\Delta BFE) + 1/2\times [4\times ar(\Delta BDE)]\\ = ar(\Delta BFE) + 2\times ar(\Delta BDE)
= 2\times ar (\Delta FED) + 2\times [ar(\Delta BFE) + ar(\Delta FED)] .....(from part (v) ar( \Delta BFE) = 2. ar( \Delta FED) ]
\\=2ar(\Delta FED) + 2[3\times ar(\Delta FED)]\\ =8\times ar(\Delta FED)

ar (FED) = \frac{1}{8} ar (AFC)
Hence proved.

Q6 Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that ar (APB) \times ar (CPD) = ar (APD) \times ar (BPC).

[ Hint: From A and C, draw perpendiculars to BD.]

Answer:

15958707858231595870782539
Given that,
A quadrilateral ABCD such that it's diagonal AC and BD intersect at P. Draw AM \perp BD and CN \perp BD

Now, ar( \Delta APB) = \frac{1}{2}\times BP\times AM and,
ar(\Delta CDP)=\frac{1}{2}\times DP\times CN
Therefore, ar( \Delta APB) \times ar( \Delta CDP)=
\\=(\frac{1}{2}\times BP\times AM).(\frac{1}{2}\times DP \times CN)\\ =\frac{1}{4}\times BP \times DP\times AM\times CN ....................(i)

Similarly, ar( \Delta APD) \times ar ( \Delta BPC) =
=\frac{1}{4}\times BP \times DP\times AM\times CN ................(ii)

From eq (i) and eq (ii), we get

ar (APB) \times ar (CPD) = ar (APD) \times ar (BPC).
Hence proved.

Q7 (i) P and Q are respectively the mid-points of sides AB and BC of a triangle ABC and R is the mid-point of AP, show that

\small ar(PRQ)=\frac{1}{2}ar(ARC)


Answer:

15958708428571595870840637
We have \Delta 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 \Delta APC,
Since R is the midpoint. So, RC is the median of the \Delta APC
Therefore, ar( \Delta ARC) = 1/2 . ar ( \Delta APC)............(i)

Also, in \Delta ABC, P is the midpoint. Thus CP is the median.
Therefore, ar( \Delta APC) = 1/2. ar ( \Delta ABC)............(ii)

Also, AQ is the median of \Delta ABC
Therefore, 1/2. ar ( \Delta ABC) = ar (ABQ)............(iii)

In \Delta APQ, RQ is the median.
Therefore, ar ( \Delta PRQ) = 1/2 .ar ( \Delta APQ).............(iv)

In \Delta ABQ, PQ is the median
Therefore, ar( \Delta APQ) = 1/2. ar( \Delta ABQ).........(v)


From eq (i),
\frac{1}{2}ar(\Delta ARC)= \frac{1}{4}ar(\Delta APC) ...........(vi)
Now, put the value of ar( \Delta APC) from eq (ii), we get
Taking RHS;
= \frac{1}{8}[ar(\Delta ABC)]
= \frac{1}{4}[ar(\Delta ABQ)] (from equation (iii))

= \frac{1}{2}[ar(\Delta APQ)] (from equation (v))

=ar(\Delta PRQ) (from equation (iv))

\Rightarrow \small ar(PRQ)=\frac{1}{2}ar(ARC)

Hence proved.

Q7 (ii) P and Q are respectively the mid-points of sides AB and BC of a triangle ABC and R is the mid-point of AP, show that
\small ar(RQC)=\frac{3}{8}ar(ABC)

Answer:

15958708500221595870849049
In \Delta RBC, RQ is the median
Therefore ar( \Delta RQC) = ar( \Delta RBQ)
= ar (PRQ) + ar (BPQ)
= 1/8 (ar \Delta ABC) + ar( \Delta BPQ) [from eq (vi) & eq (A) in part (i)]
= 1/8 (ar \Delta ABC) + 1/2 (ar \Delta PBC) [ since PQ is the median of \Delta BPC]
= 1/8 (ar \Delta ABC) + (1/2).(1/2)(ar \Delta ABC) [CP is the medain of \Delta ABC]
= 3/8 (ar \Delta ABC)

Hence proved.

Q7 (iii) P and Q are respectively the mid-points of sides AB and BC of a triangle ABC and R is the mid-point of AP, show that

ar (PBQ) = ar (ARC)

Answer:

15958708595231595870857364

QP is the median of \Delta ABQ
Therefore, ar( \Delta PBQ) = 1/2. (ar \Delta ABQ)

= (1/2). (1/2) (ar \Delta ABC) [since AQ is the median of \Delta ABC
= 1/4 (ar \Delta ABC)
= ar ( \Delta ARC) [from eq (A) of part (i)]

Hence proved.

Q8 (i) In Fig. \small 9.34 , ABC is a right triangle right angled at A. BCED, ACFG and ABMN are squares on the sides BC, CA and AB respectively. Line segment \small AX\perp DE meets BC at Y. Show that: \small \Delta MBC\cong \Delta ABD

1640236844952

Answer:

15958709530011595870949664
We have, a \Delta ABC such that BCED, ACFG and ABMN are squares on its side BC, CA and AB respec. Line segment \small AX\perp DE meets BC at Y

(i) \angle CBD = \angle MBA [each 90]
Adding \angle ABC on both sides, we get
\angle ABC + \angle CBD = \angle MBA + \angle ABC
\Rightarrow \angle ABD = \angle MBC
In \Delta ABD and \Delta MBC, we have
AB = MB
BD = BC
\angle ABD = \angle MBC
Therefore, By SAS congruency
\Delta ABD \cong \Delta MBC

Hence proved.

Q8 (ii) In Fig. \small 9.34 , ABC is a right triangle right angled at A. BCED, ACFG and ABMN are squares on the sides BC, CA and AB respectively. Line segment \small AX\perp DE meets BC at Y. Show that: \small ar(BYXD)=2ar(MBC)

Answer:

15958709644551595870961735
SInce ||gm BYXD and \Delta ABD are on the same base BD and between same parallels BD and AX
Therefore, ar( \Delta ABD) = 1/2. ar(||gm BYXD)..........(i)

But, \Delta ABD \cong \Delta MBC (proved in 1st part)
Since congruent triangles have equal areas.
Therefore, By using equation (i) we get
\small ar(BYXD)=2ar(MBC)
Hence proved.

Q8 (iii) In Fig. \small 9.34 , ABC is a right triangle right angled at A. BCED, ACFG and ABMN are squares on the sides BC, CA and AB respectively. Line segment \small AX\perp DE meets BC at Y. Show that: ar(BYXD)=ar(ABMN)

Answer:

15958709721991595870971040
Since, ar (||gm BYXD) = 2 .ar ( \Delta MBC) ..........(i) [already proved in 2nd part]
and, ar (sq. ABMN) = 2. ar ( \Delta 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

\small ar(BYXD)=ar(ABMN)

Q8 (iv) In Fig. \small 9.34 , ABC is a right triangle right angled at A. BCED, ACFG and ABMN are squares on the sides BC, CA and AB respectively. Line segment \small AX\perp DE meets BC at Y. Show that: \small \Delta FCB\cong \Delta ACE

Answer:

15958709809471595870978105
\angle FCA = \angle BCE [both 90]
By adding \angle ABC on both sides we get
\angle ABC + \angle FCA = \angle ABC + \angle BCE
\angle FCB = \angle ACE

In \Delta FCB and \Delta ACE
FC = AC [sides of square]
BC = AC [sides of square]
\angle FCB = \angle ACE
\Rightarrow
\Delta FCB \cong \Delta ACE

Hence proved.

Q8 (v) In Fig. \small 9.34 , ABC is a right triangle right angled at A. BCED, ACFG and ABMN are squares on the sides BC, CA and AB respectively. Line segment \small AX\perp DE meets BC at Y. Show that: \small ar(CYXE)=2ar(FCB)

Answer:

15958709885241595870987887
Since ||gm CYXE and \Delta ACE lie on the same base CE and between the same parallels CE and AX.
Therefore, 2. ar( \Delta ACE) = ar (||gm CYXE)
B
ut, \Delta FCB \cong \Delta ACE (in iv part)
Since the congruent triangle has equal areas. So,
\small ar(CYXE)=2ar(FCB)
Hence proved.

Q8 (vi) In Fig. \small 9.34 , ABC is a right triangle right angled at A. BCED, ACFG and ABMN are squares on the sides BC, CA and AB respectively. Line segment \small AX\perp DE meets BC at Y. Show that: \small ar(CYXE)=ar(ACFG)

Answer:

15958710006701595870997995
Since ar( ||gm CYXE) = 2. ar( \Delta ACE) {in part (v)}...................(i)
Also, \Delta FCB and quadrilateral ACFG lie on the same base FC and between the same parallels FC and BG.
Therefore, ar (quad. ACFG) = 2.ar( \Delta FCB ) ................(ii)

From eq (i) and eq (ii), we get

\small ar(CYXE)=ar(ACFG)
Hence proved.

Q8 (vii) In Fig. \small 9.34 , ABC is a right triangle right angled at A. BCED, ACFG and ABMN are squares on the sides BC, CA and AB respectively. Line segment \small AX\perp DE meets BC at Y. Show that: \small ar(BCED)=ar(ABMN)+ar(ACFG)

Answer:

15958710082891595871007517
We have,
ar(quad. BCDE) = ar(quad, CYXE) + ar(quad. BYXD)

= ar(quad, CYXE) + ar (quad. ABMN) [already proved in part (iii)]
Thus, \small ar(BCED)=ar(ABMN)+ar(ACFG)

Hence proved.

More About NCERT Solutions for Class 9 Maths Exercise 9.4

NCERT solutions Class 9 Maths exercise 9.4 includes some important concepts from the previous exercises that may prove to be crucial in order to solve some miscellanies problems from NCERT solutions Class 9 Maths exercise 9.4 we get to know another idea which manages the opposite of the above theorem. We get to realize that in case two triangles or parallelograms have a similar base (or on the other hand in case they have an equivalent base) and in case they have the equivalent area then they lie on the same parallel line.

Apart from all the above-mentioned concepts, the vice versa of these concepts (theorem) are very critical in solving some questions.

Also Read| Areas Of Parallelograms And Triangles Class 9 Notes

Benefits of NCERT Solutions for Class 9 Maths Exercise 9.4

  • Exercise 9.4 Class 9 Maths, is based on AREAS OF PARALLELOGRAMS AND TRIANGLES and most of its crucial properties.

  • From Class 9 Maths chapter 9 exercise 9.4 we get to revise the entire concepts of this chapter in a single exercise.

  • Understanding the concepts from Class 9 Maths chapter 9 exercise 9.4 will make the concepts and questions from higher standards (like Class10) easier for us.

Also, See

NCERT Solutions of Class 10 Subject Wise

Subject Wise NCERT Exemplar Solutions

Frequently Asked Questions (FAQs)

1. What is the total fundamental concept of NCERT solutions for Class 9 Maths exercise 9.4?

This exercise deals with all the difficult questions regarding the area of triangles and parallelograms and their properties.

2. If two figures A and B are congruent to each other then what can you describe between the relation of their areas?

If figure A and figure B are congruent to each other then the relation between their areas is that they are equal to each other.

Area of (A) = Area of (B)

3. What do you mean by the term “two figures lies on the same base”?

The term “two figures on the same base” means that two figures which are said to be on the same base have one common side as their base or their commo side (base length) is equal to one other.

4. Find the area of a triangle with base as 6cm and height as 4cm?

The term “two figures on the same base” means that two figures which are said to be on the same base have one common side as their base or their commo side (base length) is equal to one other.

5. Can we consider rectangle and a square both as a parallelogram?

Yes, we can consider a rectangle and a square as a parallelogram as both of its opposite sides are parallel to each other.

6. Find the area of a parallelogram with base as 12cm and height as 10cm?

Yes, we can consider a rectangle and a square as a parallelogram as both of its opposite sides are parallel to each other.

7. Name the different types of parallelograms?

The different types of parallelograms are:

  • Square

  • Rectangle 

  • Rhombus and

  • Parallelogram

8. What do you mean by figures on a similar base and between same parallels ?

Two figures are supposed to be on a same base and between same parallels if the two of them have one common (normal) side as their base and the vertices which are inverse to the normal base lie on a line parallel to the base.

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