NCERT Solutions for Exercise 9.3 Class 9 Maths Chapter 9 - Areas Of Parallelograms And Triangles

Areas Of Parallelograms And Triangles Class 9 Chapter 9 Exercise: 9.3

Q1 In Fig. $9.23$ , E is any point on median AD of a $\mathrm{\Delta }ABC$ . Show that. $ar(ABE)=ar(ACE)$

Answer:
We have $\mathrm{\Delta }$ ABC such that AD is a median. And we know that median divides the triangle into two triangles of equal areas.
Therefore, ar( $\mathrm{\Delta }$ ABD) = ar( $\mathrm{\Delta }$ ACD)............(i)

Similarly, In triangle $\mathrm{\Delta }$ BEC,
ar( $\mathrm{\Delta }$ BED) = ar ( $\mathrm{\Delta }$ DEC)................(ii)

On subtracting eq(ii) from eq(i), we get
ar( $\mathrm{\Delta }$ ABD) - ar( $\mathrm{\Delta }$ BED) =
$ar(ABE)=ar(ACE)$

Hence proved.

Q2 In a triangle ABC, E is the mid-point of median AD. Show that $ar(BED)=\frac{1}{4}ar(ABC)$ .

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.
$\therefore$ ar( $\mathrm{\Delta }$ ABD) = ar ( $\mathrm{\Delta }$ ACD) = 1/2 ar( $\mathrm{\Delta }$ ABC)..............(i)
Now, in triangle $\mathrm{\Delta }$ ABD,
BE is the median [since E is the midpoint of AD]
$\therefore$ ar ( $\mathrm{\Delta }$ BED) = 1/2 ar( $\mathrm{\Delta }$ ABD)........(ii)

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

ar ( $\mathrm{\Delta }$ BED) = 1/2 . (1/2 ar(ar ( $\mathrm{\Delta }$ ABC))
ar ( $\mathrm{\Delta }$ BED) = 1/4 .ar( $\mathrm{\Delta }$ 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 ( $\mathrm{\Delta }$ BOC) = ar ( $\mathrm{\Delta }$ DOC)...........(a) ( since OC is the median of triangle CBD)

Similarly, ar( $\mathrm{\Delta }$ AOD) = ar( $\mathrm{\Delta }$ DOC) ............(b) ( since OD is the median of triangle ACD)

and, ar ( $\mathrm{\Delta }$ AOB) = ar( $\mathrm{\Delta }$ BOC)..............(c) ( since OB is the median of triangle ABC)

From eq (a), (b) and eq (c), we get

ar ( $\mathrm{\Delta }$ BOC) = ar ( $\mathrm{\Delta }$ DOC)= ar( $\mathrm{\Delta }$ AOD) = ( $\mathrm{\Delta }$ AOB)

Thus, the diagonals of ||gm divide it into four equal triangles of equal area.

Q4 In Fig. $9.24$ , ABC and ABD are two triangles on the same base AB. If line- segment CD is bisected by AB at O, show that $ar(ABC)=ar(ABD)$ .

Answer:

We have, $\mathrm{\Delta }$ ABC and $\mathrm{\Delta }$ ABD on the same base AB. CD is bisected by AB at point O.
$\therefore$ OC = OD
Now, in $\mathrm{\Delta }$ ACD, AO is median
$\therefore$ ar ( $\mathrm{\Delta }$ AOC) = ar ( $\mathrm{\Delta }$ AOD)..........(i)

Similarly, in $\mathrm{\Delta }$ BCD, BO is the median
$\therefore$ ar ( $\mathrm{\Delta }$ BOC) = ar ( $\mathrm{\Delta }$ BOD)............(ii)

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

ar ( $\mathrm{\Delta }$ AOC) + ar ( $\mathrm{\Delta }$ BOC) = ar ( $\mathrm{\Delta }$ AOD) + ar ( $\mathrm{\Delta }$ BOD)

$ar(ABC)=ar(ABD)$

Hence proved.

Q5 (i) D, E and F are respectively the mid-points of the sides BC, CA and AB of a $\mathrm{\Delta }ABC$ . Show that BDEF is a parallelogram.

Answer:

We have a triangle $\mathrm{\Delta }$ ABC such that D, E and F are the midpoints of the sides BC, CA and AB respectively.

Now, in $\mathrm{\Delta }$ 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.
$\therefore$ EF || BC or EF || BD

also, EF = 1/2 (BC)
$\Rightarrow EF=BD$ [ 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 $\mathrm{\Delta }ABC$ . Show that

$ar(DEF)=\frac{1}{4}ar(ABC)$

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
$\therefore$ Ar (BDEF) = Ar (DCEF)
$\Rightarrow$ Ar( $\mathrm{\Delta }$ BDF) = Ar ( $\mathrm{\Delta }$ DEF) .............(i)

It is known that diagonals of ||gm divides it into two triangles of equal area.
$\therefore$ Ar(DCE) = Ar (DEF).......(ii)

and, Ar( $\mathrm{\Delta }$ AEF) = Ar ( $\mathrm{\Delta }$ DEF)...........(iii)

From equation(i), (ii) and (iii), we get

Ar( $\mathrm{\Delta }$ BDF) = Ar(DCE) = Ar( $\mathrm{\Delta }$ AEF) = Ar ( $\mathrm{\Delta }$ DEF)

Thus, Ar ( $\mathrm{\Delta }$ ABC) = Ar( $\mathrm{\Delta }$ BDF) + Ar(DCE) + Ar( $\mathrm{\Delta }$ AEF) + Ar ( $\mathrm{\Delta }$ DEF)
Ar ( $\mathrm{\Delta }$ ABC) = 4 . Ar( $\mathrm{\Delta }$ DEF)
$\Rightarrow$ $ar(\mathrm{\Delta }DEF)=\frac{1}{4}ar(\mathrm{\Delta }ABC)$

Hence proved.

Q5 (iii) D, E and F are respectively the mid-points of the sides BC, CA and AB of a $\mathrm{\Delta }ABC$ . Show that

$ar(BDEF)=\frac{1}{2}ar(ABC)$

Answer:

Since we already proved that,
ar( $\mathrm{\Delta }$ DEF) = ar ( $\mathrm{\Delta }$ BDF).........(i)

So, ar(||gm BDEF) = ar( $\mathrm{\Delta }$ BDF) + ar ( $\mathrm{\Delta }$ DEF)
= 2 . ar( $\mathrm{\Delta }$ DEF) [from equation (i)]
$$\begin{gathered} =2[\frac{1}{4}ar(\mathrm{\Delta }ABC)] \\ =\frac{1}{2}ar(\mathrm{\Delta }ABC) \end{gathered}$$

Hence proved.

Q6 (i) In Fig. $9.25$ , diagonals AC and BD of quadrilateral ABCD intersect at O such that. If $AB=CD$ , then show that:

$ar(DOC)=ar(AOB)$

[ 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 $\perp$ AC and FB $\perp$ AC

In $\mathrm{\Delta }$ DEO and $\mathrm{\Delta }$ BFO
$\mathrm{\angle }$ DOE = $\mathrm{\angle }$ BOF [vertically opposite angle]
$\mathrm{\angle }$ OED = $\mathrm{\angle }$ BFO [each ${90}^0$ ]
OB = OD [given]

Therefore, by AAS congruency
$\mathrm{\Delta }$ DEO $\cong$ $\mathrm{\Delta }$ BFO
$\Rightarrow$ DE = FB [by CPCT]

and ar( $\mathrm{\Delta }$ DEO) = ar( $\mathrm{\Delta }$ BFO) ............(i)

Now, In $\mathrm{\Delta }$ DEC and $\mathrm{\Delta }$ ABF
$\mathrm{\angle }$ DEC = $\mathrm{\angle }$ BFA [ each ${90}^0$ ]
DE = FB
DC = BA [given]
So, by RHS congruency
$\mathrm{\Delta }$ DEC $\cong$ $\mathrm{\Delta }$ BFA
$\mathrm{\angle }$ 1 = $\mathrm{\angle }$ 2 [by CPCT]
and, ar( $\mathrm{\Delta }$ DEC) = ar( $\mathrm{\Delta }$ BFA).....(ii)

By adding equation(i) and (ii), we get
$ar(DOC)=ar(AOB)$
Hence proved.

Q6 (ii) In Fig. $9.25$ , diagonals AC and BD of quadrilateral ABCD intersect at O such that $OB=OD$ . If $AB=CD$ , then show that: $ar(DCB)=ar(ACB)$

[ Hint: From D and B, draw perpendiculars to AC.]

Answer:

We already proved that,
$ar(\mathrm{\Delta }DOC)=ar(\mathrm{\Delta }AOB)$
Now, add $ar(\mathrm{\Delta }BOC)$ on both sides we get

$$\begin{gathered} ar(\mathrm{\Delta }DOC)+ar(\mathrm{\Delta }BOC)=ar(\mathrm{\Delta }AOB)+ar(\mathrm{\Delta }BOC) \\ ar(\mathrm{\Delta }DCB)=ar(\mathrm{\Delta }ACB) \end{gathered}$$
Hence proved.

Q6 (iii) In Fig. $9.25$ , diagonals AC and BD of quadrilateral ABCD intersect at O such that $OB=OD$ . If $AB=CD$ , then show that:

$DA\parallel CB$ or ABCD is a parallelogram.

[ Hint : From D and B, draw perpendiculars to AC.]

Answer:

Since $\mathrm{\Delta }$ DCB and $\mathrm{\Delta }$ ACB both lie on the same base BC and having equal areas.
Therefore, They lie between the same parallels BC and AD
$\Rightarrow$ CB || AD
also $\mathrm{\angle }$ 1 = $\mathrm{\angle }$ 2 [ already proved]
So, AB || CD
Hence ABCD is a || gm

Q7 D and E are points on sides AB and AC respectively of $\mathrm{\Delta }ABC$ such that $ar(DBC)=ar(EBC)$ . Prove that $DE\parallel BC$ .

Answer:

We have $\mathrm{\Delta }$ ABC and points D and E are on the sides AB and AC such that ar( $\mathrm{\Delta }$ DBC ) = ar ( $\mathrm{\Delta }$ EBC)


Since $\mathrm{\Delta }$ DBC and $\mathrm{\Delta }$ EBC are on the same base BC and having the same area.
$\therefore$ They must lie between the same parallels DE and BC
Hence DE || BC.

Q8 XY is a line parallel to side BC of a triangle ABC. If $BE\parallel AC$ and $CF\parallel AB$ meet XY at E and F respectively, show that $ar(ABE)=ar(ACF)$

Answer:

We have a $\mathrm{\Delta }$ ABC such that BE || AC and CF || AB
Since XY || BC and BE || CY
Therefore, BCYE is a ||gm
Now, The ||gm BCEY and $\mathrm{\Delta }$ ABE are on the same base BE and between the same parallels AC and BE.
$\therefore$ ar( $\mathrm{\Delta }$ AEB) = 1/2 .ar(||gm BEYC)..........(i)
Similarly, ar( $\mathrm{\Delta }$ 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.
$\therefore$ ar (BEYC) = ar (BCFX).........(iii)

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

ar( $\mathrm{\Delta }$ ABE) = ar( $\mathrm{\Delta }$ ACF)
Hence proved.

Q9 The side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed (see Fig. $9.26$ ). Show that $ar(ABCD)=ar(PBQR)$ . [ Hint: Join AC and PQ. Now compare $ar(ACQ)$ and $ar(APQ)$ .]

Answer:

Join the AC and PQ.

It is given that ABCD is a ||gm and AC is a diagonal of ||gm
Therefore, ar( $\mathrm{\Delta }$ ABC) = ar( $\mathrm{\Delta }$ ADC) = 1/2 ar(||gm ABCD).............(i)

Also, ar( $\mathrm{\Delta }$ PQR) = ar( $\mathrm{\Delta }$ BPQ) = 1/2 ar(||gm PBQR).............(ii)

Since $\mathrm{\Delta }$ AQC and $\mathrm{\Delta }$ APQ are on the same base AQ and between same parallels AQ and CP.
$\therefore$ ar( $\mathrm{\Delta }$ AQC) = ar ( $\mathrm{\Delta }$ APQ)

Now, subtracting $\mathrm{\Delta }$ ABQ from both sides we get,

ar( $\mathrm{\Delta }$ AQC) - ar ( $\mathrm{\Delta }$ ABQ) = ar ( $\mathrm{\Delta }$ APQ) - ar ( $\mathrm{\Delta }$ ABQ)
ar( $\mathrm{\Delta }$ ABC) = ar ( $\mathrm{\Delta }$ BPQ)............(iii)

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

$ar(ABCD)=ar(PBQR)$

Hence proved.

Q10 Diagonals AC and BD of a trapezium ABCD with $AB\parallel DC$ intersect each other at O. Prove that $ar(AOD)=ar(BOC)$ .

Answer:

We have a trapezium ABCD such that AB || CD and it's diagonals AC and BD intersect each other at O
$\mathrm{\Delta }$ ABD and $\mathrm{\Delta }$ ABC are on the same base AB and between same parallels AB and CD
$\therefore$ ar( $\mathrm{\Delta }$ ABD) = ar ( $\mathrm{\Delta }$ ABC)

Now, subtracting $\mathrm{\Delta }$ AOB from both sides we get

ar ( $\mathrm{\Delta }$ AOD) = ar ( $\mathrm{\Delta }$ BOC)

Hence proved.

Q11 In Fig. $9.27$ , ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show that

(i) $ar(ACB)=ar(ACF)$
(ii) $ar(AEDF)=ar(ABCDE)$

Answer:
We have a pentagon ABCDE in which BF || AC and CD is produced to F.

(i) Since $\mathrm{\Delta }$ ACB and $\mathrm{\Delta }$ ACF are on the same base AC and between same parallels AC and FB.
$\therefore$ ar( $\mathrm{\Delta }$ ACB) = ar ( $\mathrm{\Delta }$ ACF)..................(i)

(ii) Adding the ar (AEDC) on both sides in equation (i), we get

ar( $\mathrm{\Delta }$ ACB) + ar(AEDC) = ar ( $\mathrm{\Delta }$ ACF) + ar(AEDC)
$\therefore$ $ar(AEDF)=ar(ABCDE)$

Hence proved.

Q12 A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decided to take over some portion of his plot from one of the corners to construct a Health Centre. Itwaari agrees to the above proposal with the condition that he should be given an equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot. Explain how this proposal will be implemented.

Answer:

We have a quadrilateral shaped plot ABCD. Draw DF || AC and AF || CF.
Now, $\mathrm{\Delta }$ DAF and $\mathrm{\Delta }$ DCF are on the same base DF and between same parallels AC and DF.
$\therefore$ ar ( $\mathrm{\Delta }$ DAF) = ar( $\mathrm{\Delta }$ DCF)
On subtracting $\mathrm{\Delta }$ DEF from both sides, we get

ar( $\mathrm{\Delta }$ ADE) = ar( $\mathrm{\Delta }$ CEF)...............(i)
The portion of $\mathrm{\Delta }$ ADE can be taken by the gram panchayat and on adding the land $\mathrm{\Delta }$ CEF to his (Itwaari) land so as to form a triangular plot.( $\mathrm{\Delta }$ ABF)

We need to prove that ar( $\mathrm{\Delta }$ ABF) = ar (quad. ABCD)

Now, adding ar(quad. ABCE) on both sides in eq (i), we get

ar ( $\mathrm{\Delta }$ ADE) + ar(quad. ABCE) = ar( $\mathrm{\Delta }$ CEF) + ar(quad. ABCE)
ar (ABCD) = ar( $\mathrm{\Delta }$ ABF)

Q13 ABCD is a trapezium with $AB\parallel DC$ . A line parallel to AC intersects AB at X and BC at Y. Prove that $ar(ADX)=ar(ACY)$ . [ Hint: Join CX.]

Answer:

We have a trapezium ABCD, AB || CD
XY ||AC meets AB at X and BC at Y. Join XC

Since $\mathrm{\Delta }$ ADX and $\mathrm{\Delta }$ ACX lie on the same base CD and between same parallels AX and CD
Therefore, ar( $\mathrm{\Delta }$ ADX) = ar( $\mathrm{\Delta }$ ACX)..........(i)
Similarly ar( $\mathrm{\Delta }$ ACX) = ar( $\mathrm{\Delta }$ ACY).............(ii) [common base AC and AC || XY]
From eq (i) and eq (ii), we get

ar( $\mathrm{\Delta }$ ADX) = ar ( $\mathrm{\Delta }$ ACY)

Hence proved.

Q14 In Fig. $9.28$ , $AP\parallel BQ\parallel CR$ . Prove that $ar(AQC)=ar(PBR)$ .

Answer:

We have, AP || BQ || CR
$\mathrm{\Delta }$ BCQ and $\mathrm{\Delta }$ BQR lie on the same base (BQ) and between same parallels (BQ and CR)
Therefore, ar ( $\mathrm{\Delta }$ BCQ) = ar ( $\mathrm{\Delta }$ BQR)........(i)

Similarly, ar ( $\mathrm{\Delta }$ ABQ) = ar ( $\mathrm{\Delta }$ PBQ) [common base BQ and BQ || AP]............(ii)

Add the eq(i) and (ii), we get

ar ( $\mathrm{\Delta }$ AQC) = ar ( $\mathrm{\Delta }$ PBR)

Hence proved.

Q15 Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that $ar(AOD)=ar(BOC)$ . Prove that ABCD is a trapezium.

Answer:

We have,
ABCD is a quadrilateral and diagonals AC and BD intersect at O such that ar( $\mathrm{\Delta }$ AOD) = ar ( $\mathrm{\Delta }$ BOC) ...........(i)

Now, add ar ( $\mathrm{\Delta }$ BOA) on both sides, we get

ar( $\mathrm{\Delta }$ AOD) + ar ( $\mathrm{\Delta }$ BOA) = ar ( $\mathrm{\Delta }$ BOA) + ar ( $\mathrm{\Delta }$ BOC)
ar ( $\mathrm{\Delta }$ ABD) = ar ( $\mathrm{\Delta }$ ABC)
Since the $\mathrm{\Delta }$ ABC and $\mathrm{\Delta }$ ABD lie on the same base AB and have an equal area.
Therefore, AB || CD

Hence ABCD is a trapezium.

Q16 In Fig. $9.29$ , $ar(DRC)=ar(DPC)$ and $ar(BDP)=ar(ARC)$ . Show that both the quadrilaterals ABCD and DCPR are trapeziums.

Answer:

Given,
ar( $\mathrm{\Delta }$ DPC) = ar( $\mathrm{\Delta }$ DRC) ..........(i)
and ar( $\mathrm{\Delta }$ BDP) = ar(ARC)............(ii)

from equation (i),
Since $\mathrm{\Delta }$ DRC and $\mathrm{\Delta }$ DPC lie on the same base DC and between same parallels.
$\therefore$ CD || RP (opposites sides are parallel)

Hence quadrilateral DCPR is a trapezium

Now, by subtracting eq(ii) - eq(i) we get

ar( $\mathrm{\Delta }$ BDP) - ar( $\mathrm{\Delta }$ DPC) = ar( $\mathrm{\Delta }$ ARC) - ar( $\mathrm{\Delta }$ DRC)
ar( $\mathrm{\Delta }$ BDC) = ar( $\mathrm{\Delta }$ ADC) (Since theya are on the same base DC)
$\Rightarrow$ AB || DC

Hence ABCD is a trapezium.