ICSE Class 9 Area Theorems Solution New Pattern By Clarify Knowledge

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ICSE Class 9 Area Theorems Solution Solution Table

Chapter 16 - Area Theorems [Proof and Use] Exercise Ex. 16(B)

Question 1

Show that:

(i) A diagonal divides a parallelogram into two triangles of equal area.

(ii) The ratio of the areas of two triangles of the same height is equal to the ratio of their bases.

(iii) The ratio of the areas of two triangles on the same base is equal to the ratio of their heights.Solution 1

(i) Suppose ABCD is a parallelogram (given)

Area of congruent triangles are equal.

Therefore, Area of ABC = Area of ADC

(ii) Consider the following figure:

Here 

Since Ar.()=

And, Ar.()=

,

hence proved

(iii) Consider the following figure:

Here

Ar.()=

And, Ar.()=

,

hence provedQuestion 2

In the given figure; AD is median of ABC and E is any point on median AD. Prove that Area (ABE) = Area (ACE).

Solution 2

AD is the median of ABC. Therefore it will divide ABC into two triangles of equal areas.

Area(ABD)= Area(ACD)   (i)

ED is the median of EBC

Area(EBD)= Area(ECD)  (ii)

Subtracting equation (ii) from (i), we obtain

Area(ABD)- Area(EBD)= Area(ACD)- Area(ECD)

Area (ABE) = Area (ACE). Hence provedQuestion 3

In the figure of question 2, if E is the mid point of median AD, then prove that:

Area (ABE) = Area (ABC).Solution 3

AD is the median of ABC. Therefore it will divide ABC into two triangles of equal areas.

Area(ABD)= Area(ACD)

Area (ABD)= Area(ABC) (i)

In ABD, E is the mid-point of AD. Therefore BE is the median.

Area(BED)= Area(ABE)

Area(BED)= Area(ABD)

Area(BED)= Area(ABC)[from equation (i)]

Area(BED)= Area(ABC)Question 4

ABCD is a parallelogram. P and Q are the mid-points of sides AB and AD respectively.

Prove that area of triangle APQ = of the area of parallelogram ABCD.Solution 4

We have to join PD and BD.

BD is the diagonal of the parallelogram ABCD. Therefore it divides the parallelogram into two equal parts.

Area(ABD)= Area(DBC)

=Area (parallelogram ABCD) (i)

DP is the median of ABD. Therefore it will divide ABD into two triangles of equal areas.

Area(APD)= Area(DPB)

= Area (ABD)

= Area(parallelogram ABCD)[from equation (i)]

= Area (parallelogram ABCD) (ii)

In APD, Q is the mid-point of AD. Therefore PQ is the median.

Area(APQ)= Area(DPQ)

= Area(APD)

= Area (parallelogram ABCD) [from equation (ii)]

Area (APQ)= Area (parallelogram ABCD),hence provedQuestion 5

The base BC of triangle ABC is divided at D so that BD = DC.

Prove that area of ABD = of the area of ABC.Solution 5

In ABC, BD = DC

Ar.(ABD):Ar.(ADC)=1:2

But Ar.(ABD)+Ar.(ADC)=Ar.(ABC)

Ar.(ABD)+2Ar.(ABD)=Ar.(ABC)

3 Ar.(ABD)= Ar.(ABC)

Ar.(ABD)= Ar.(ABC)Question 6

In a parallelogram ABCD, point P lies in DC such that DP: PC = 3:2. If area of ΔDPB = 30 sq. cm, find the area of the parallelogram ABCD.Solution 6

Ratio of area of triangles with same vertex and bases along the same line is equal to ratio of their respective bases. So, we have

Given: Area of ΔDPB = 30 sq. cm

So area of ΔPCB = 20 sq. cm

Consider the following figure.

From the diagram, it is clear that,

Diagonal of the parallelogram divides it into two triangles  of equal area.

Question 7

ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F.

If ar.(∆DFB) = 30 cm²; find the area of parallelogramSolution 7

Question 8

The following figure shows a triangle ABC in which P, Q and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ:

Prove that: ar.(∆ ABC) = 8 × ar.(∆ QSB)

Solution 8

Chapter 16 - Area Theorems [Proof and Use] Exercise Ex. 16(C)

Question 1

In the given figure, the diagonals AC and BD intersect at point O. If OB = OD and AB//DC, show that:

(i) Area (Δ DOC) = Area (Δ AOB).

(ii) Area (Δ DCB) = Area (Δ ACB).

(iii) ABCD is a parallelogram.

Solution 1

(i)

Ratio of area of triangles with same vertex and bases along the same line is equal to the ratio of their respective bases. So, we have:

 ----1

Similarly

 ------2

We know that area of triangles on the same base and between same parallel lines are equal.

Area of Δ ACD = Area of Δ BCD

Area of Δ AOD + Area of Δ DOC = Area of Δ DOC + Area of Δ BOC

=> Area of Δ AOD = Area of Δ BOC ------3

From 1, 2 and 3 we have

Area (Δ DOC) = Area (Δ AOB)

Hence Proved.

(ii)

Similarly, from 1, 2 and 3, we also have

Area of Δ DCB = Area of Δ DOC + Area of Δ BOC = Area of Δ AOB + Area of Δ BOC = Area of Δ ABC

So Area of Δ DCB = Area of Δ ABC

Hence Proved.

(iii)

We know that area of triangles on the same base and between same parallel lines are equal.

Given: triangles are equal in area on the common base, so it indicates AD|| BC.

So, ABCD is a parallelogram.

Hence ProvedQuestion 2

The given figure shows a parallelogram ABCD with area 324 sq. cm. P is a point in AB such that AP:PB = 1:2 Find The area of Δ APD.

Solution 2

Ratio of area of triangles with the same vertex and bases along the same line is equal to the ratio of their respective bases.

So, we have

Area of parallelogram ABCD = 324 sq.cm

Area of the triangles with the same base and between the same parallels are equal.

We know that area of the triangle is half the area of the parallelogram if they lie on the same base and between the

parallels.

Therefore, we have,

(ii)

Hence OP:OD = 1:3Question 3

In ABC, E and F are mid-points of sides AB and AC respectively. If BF and CE intersect each other at point O, prove that the OBC and quadrilateral AEOF are equal in area.Solution 3

E and F are the midpoints of the sides AB and AC.

Consider the following figure.

Therefore, by midpoint theorem, we have, EF || BC

Triangles BEF and CEF lie on the common base EF and between the parallels, EF and BC

Now BF and CE are the medians of the triangle ABC

Medians of the triangle divides it into two equal areas of triangles.

Thus, we have, Ar.ABF=Ar.CBF

Subtracting Ar.BOE on the both the sides, we have

Ar.ABF - Ar.BOE=Ar.CBF - Ar.BOE

Since, Ar.(BOE)= Ar.(COF),

Ar.ABF- Ar.BOE=Ar.CBF- Ar.COF

Ar. (quad. AEOF)=Ar.(OBC), hence provedQuestion 4

In parallelogram ABCD, P is mid-point of AB. CP and BD intersect each other at point O. If area of POB = 40 cm², and OP : OC = 1:2, find:

(i) Areas of BOC and PBC

(ii) Areas of ABC and parallelogram ABCD.Solution 4

Question 5

The medians of a triangle ABC intersect each other at point G. If one of its medians is AD, prove that:

(i) Area (ABD) = 3 Area (BGD)

(ii) Area (ACD) = 3 Area (CGD)

(iii) Area (BGC) = Area (ABC)Solution 5

(i) The figure is shown below

(ii)

(iii)

Question 6

The perimeter of a triangle ABC is 37 cm and the ratio between the lengths of its altitudes be 6 : 5 : 4. Find the lengths of its sides.

Let the sides be x cm, y cm and (37 - x - y) cm. Also, let the lengths of altitudes be 6a cm, 5a cm and 4a cm.Solution 6

Consider that the sides be x cm, y cm and (37-x-y) cm. also, consider that the lengths of altitudes be 6a cm, 5a cm and 4a cm.

Area of a triangle=basealtitude

and

and

Solving both the equations, we have

X=10 cm, y=12 cm and (37-x-y)cm=15 cmQuestion 7

In parallelogram ABCD, E is a point in AB and DE meets diagonal AC at point F. If DF: FE = 5:3 and area of  ADF is

60 cm² ; find

(i) area of ADE

(ii) if AE :EB = 4:5, find the area  of  ADB

(iii) aslo, find area of parallelogram ABCDSolution 7

Question 8

Solution 8

Question 9

In the following figure, OAB is a triangle and AB∥DC.

If the area of ∆ CAD = 140 cm² and the area of ∆ ODC = 172 cm², find

(i) the area of ∆ DBC

(ii) the area of ∆ OAC

(iii) the area of ∆ ODB.Solution 9

Question 10

E, F, G and H are the mid- points of the sides of a parallelogram ABCD. Show that area of quadrilateral EFGH is half of the area of parallelogram ABCD.Solution 10

Join HF.

Since H and F are mid-points of AD and BC respectively,

Now, ABCD is a parallelogram.

⇒ AD = BC and AD ∥ BC

⇒ AH = BF and AH ∥ BF

⇒ ABFH is a parallelogram.

Since parallelogram FHAB and ΔFHE are on the same base FH and between the same parallels HF and AB,

Question 11

ABCD is a trapezium with AB parallel to DC. A line parallel to AC intersects AB at X and BC at Y. Prove that area of ∆ADX = area of ∆ACY.Solution 11

Join CX, DX and AY.

Now, triangles ADX and ACX are on the same base AX and between the parallels AB and DC.

∴ A(ΔADX) = A(ΔACX) ….(i)

Also, triangles ACX and ACY are on the same base AC and between the parallels AC and XY.

∴ A(ΔACX) = A(ΔACY) ….(ii)

From (i) and (ii), we get

A(ΔADX) = A(ΔACY)

Chapter 16 - Area Theorems [Proof and Use] Exercise Ex. 16(A)

Question 1

In the given figure, if area of triangle ADE is 60 cm², state, given reason, the area of :

(i) Parallelogram ABED;

(ii) Rectangle ABCF;

(iii) Triangle ABE.

Solution 1

(i)and parallelogram ABED are on the same base AB and between the same parallels DE//AB, so area of the triangle is half the area of parallelogram ABED.

Area of ABED = 2 (Area of ADE) = 120 cm²

(ii)Area of parallelogram is equal to the area of rectangle on the same base and of the same altitude i.e, between the same parallels

Area of ABCF = Area of ABED = 120 cm²

(iii)We know that area of triangles on the same base and between same parallel lines are equal

Area of ABE=Area of ADE =60 cm²Question 2

The given figure shows a rectangle ABDC and a parallelogram ABEF; drawn on opposite sides of AB. Prove that:

(i) Quadrilateral CDEF is a parallelogram;

(ii) Area of quad. CDEF

= Area of rect. ABDC

+ Area of // gm. ABEF.

Solution 2

After drawing the opposite sides of AB, we get

Since from the figure, we get CD//FE therefore FC must parallel to DE. Therefore it is proved that the quadrilateral CDEF is a parallelogram.

Area of parallelogram on same base and between same parallel lines is always equal and area of parallelogram is equal to the area of rectangle on the same base and of the same altitude i.e, between same parallel lines. 

So Area of CDEF= Area of ABDC + Area of ABEF 

Hence ProvedQuestion 3

In the given figure, diagonals PR and QS of the parallelogram PQRS intersect at point O and LM is parallel to PS. Show that:

(i) 2 Area (POS) = Area (// gm PMLS)

(ii) Area (POS) + Area (QOR)

= Area (// gm PQRS)

(iii) Area (POS) + Area (QOR)

= Area (POQ) + Area (SOR).Solution 3

(i)

Since POS and parallelogram PMLS are on the same base PS and between the same parallels i.e. SP//LM.

As O is the center of LM and Ratio of area of triangles with same vertex and bases along the same line is equal to ratio of their respective bases.

The area of the parallelogram is  twice the area of the triangle if they lie on the same base and in between the same parallels.

So 2(Area of PSO)=Area of PMLS

Hence Proved.

(ii)

Consider the expression :

LM is parallel to PS and PS is parallel to RQ, therefore, LM is

Since triangle POS lie on the base PS and in between the parallels PS and LM, we have,,

Since triangle QOR lie on the base QR and in between the parallels LM and RQ, we have,

(iii)

In a parallelogram, the diagonals bisect each other. 

Therefore, OS = OQ

Consider the triangle PQS, since OS = OQ, OP is the median of the triangle PQS.

We know that median of a triangle divides it into two triangles of equal area.

Therefore,

Hence Proved.Question 4

In parallelogram ABCD, P is a point on side AB and Q is a point on side BC.

Prove that:

(i) CPD and AQD are equal in area.

(ii) Area (AQD)

= Area (APD) + Area (CPB)Solution 4

 (i)

Given ABCD is a parallelogram. P and Q are any points on the sides AB and BC respectively, join diagonals AC and BD.

proof:

since triangles with same base and between same set of parallel lines have equal areas

area (CPD)=area(BCD)…… (1)

again, diagonals of the parallelogram bisects area in two equal parts

area (BCD)=(1/2) area of parallelogram ABCD…… (2)

from (1) and (2)

area(CPD)=1/2 area(ABCD)…… (3)

similarly area (AQD)=area(ABD)=1/2 area(ABCD)…… (4)

from (3) and (4)

area(CPD)=area(AQD),

hence proved.

(ii)

We know that area of triangles on the same base and between same parallel lines are equal

So Area of AQD= Area of ACD= Area of PDC = Area of BDC = Area of ABC=Area of APD + Area of BPC

Hence ProvedQuestion 5

In the given figure, M and N are the mid-points of the sides DC and AB respectively of the parallelogram ABCD.

If the area of parallelogram ABCD is 48 cm²;

(i) State the area of the triangle BEC.

(ii) Name the parallelogram which is equal in area to the triangle BEC.Solution 5

(i)

Since triangle BEC and parallelogram ABCD are on the same base BC and between the same parallels i.e. BC//AD.

(ii)

Therefore, Parallelograms ANMD and NBCM have areas equal to triangle BECQuestion 6

In the following figure, CE is drawn parallel to diagonals DB of the quadrilateral ABCD which meets AB produced at point E.

Prove that ADE and quadrilateral ABCD are equal in area.

Solution 6

Since DCB and DEB are on the same base DB and between the same parallels i.e. DB//CE, therefore we get

Hence provedQuestion 7

ABCD is a parallelogram a line through A cuts DC at point P and BC produced at Q. Prove that triangle BCP is equal in area to triangle DPQ.

Solution 7

APB and parallelogram ABCD are on the same base AB and between the same parallel lines AB and CD.

ADQ and parallelogram ABCD are on the same base AD and between the same parallel lines AD and BQ.

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

Subtracting Ar.PCQ from both sides, we get

Hence proved.Question 8

The given figure shows a pentagon ABCDE. EG drawn parallel to DA meets BA produced at G and CF draw parallel to DB meets AB produced at F.

Prove that the area of pentagon ABCDE is equal to the area of triangle GDF.

Solution 8

Since triangle EDG and EGA are on the same base EG and between the same parallel lines EG and DA, therefore

Subtracting from both sides, we have

  (i)

Similarly

  (ii)

Now

Hence provedQuestion 9

In the given figure, AP is parallel to BC, BP is parallel to CQ. Prove that the area of triangles ABC and BQP are equal.

Solution 9

Joining PC we get

ABC and BPC are on the same base BC and between the same parallel lines AP and BC.

BPC and BQP are on the same base BP and between the same parallel lines BP and CQ.

From (i) and (ii), we get

Hence proved.Question 10

In the figure given alongside, squares ABDE and AFGC are drawn on the side AB and the hypotenuse AC of the right triangle ABC.

If BH is perpendicular to FG prove that:

(i) EAC BAF.

(ii) Area of the square ABDE

= Area of the rectangle ARHF.Solution 10

(i)

From (i) and (ii), we get

In EAC and BAF, we have, EA=AB

and AC=AF

EAC BAF (SAS axiom of congruency)

(ii)

Question 11

In the following figure, DE is parallel to BC. Show that:

(i) Area (ADC) = Area(AEB).

(ii) Area (BOD) = Area (COE).

Solution 11

(i)

In ABC, D is midpoint of AB and E is the midpoint of AC.

DE is parallel to BC.

Again

From the above two equations, we have

Area (ADC) = Area(AEB).

Hence Proved

(ii)

We know that area of triangles on the same base and between same parallel lines are equal

Area(triangle DBC)= Area(triangle BCE)

Area(triangle DOB) + Area(triangle BOC) = Area(triangle BOC) + Area(triangle COE)

So Area(triangle DOB) = Area(triangle COE)Question 12

ABCD and BCFE are parallelograms. If area of triangle EBC = 480 cm²; AB = 30 cm and BC = 40 cm; Calculate;

(i) Area of parallelogram ABCD;

(ii) Area of the parallelogram BCFE;

(iii) Length of altitude from A on CD;

(iv) Area of triangle ECF.

Solution 12

(i)

Since EBC and parallelogram ABCD are on the same base BC and between the same parallels i.e. BC//AD.

(ii)

Parallelograms on same base and between same parallels are equal in area

Area of BCFE = Area of ABCD= 960 cm²

(iii)

Area of triangle ACD=480 = (1/2) x 30 x Altitude

Altitude=32 cm

(iv)

The area of a triangle is half that of a parallelogram on the same base and between the same parallels.

Therefore,

Question 13

In the given figure, D is mid-point of side AB of ABC and BDEC is a parallelogram.

Prove that:

Area of ABC = Area of // gm BDEC.Solution 13

Here AD=DB and EC=DB, therefore EC=AD

Again, (opposite angles)

Since ED and CB are parallel lines and AC cut this line, therefore

From the above conditions, we have

Adding quadrilateral CBDF in both sides, we have

Area of // gm BDEC= Area of ABCQuestion 14

In the following, AC // PS // QR and PQ // DB // SR.

Prove that:

Area of quadrilateral PQRS = 2 Area of quad. ABCD.

Solution 14

In Parallelogram PQRS, AC // PS // QR and PQ // DB // SR.

Similarly, AQRC and APSC are also parallelograms.

Since ABC and parallelogram AQRC are on the same base AC and between the same parallels, then

Ar.(ABC)=Ar.(AQRC)......(i)

Similarly,

Ar.(ADC)=Ar.(APSC).......(ii)

Adding (i) and (ii), we get

Area of quadrilateral PQRS = 2 Area of quad. ABCDQuestion 15

ABCD is trapezium with AB // DC. A line parallel to AC intersects AB at point M and BC at point N. Prove that: area of Δ ADM = area of Δ ACN.Solution 15

Given: ABCD is a trapezium

AB || CD, MN || AC

Join C and M

We know that area of triangles on the same base and between same parallel lines are equal.

So Area of Δ AMD = Area of Δ AMC

Similarly, consider AMNC quadrilateral where MN || AC.

Δ ACM and Δ ACN are on the same base and between the same parallel lines. So areas are equal.

So, Area of Δ ACM = Area of Δ CAN

From the above two equations, we can say

Area of Δ ADM = Area of Δ CAN

Hence Proved.Question 16

In the given figure, AD // BE // CF. Prove that area (ΔAEC) = area (ΔDBF)

Solution 16

We know that area of triangles on the same base and between same parallel lines are equal.

Consider ABED quadrilateral; AD||BE

With common base, BE and between AD and BE parallel lines, we have

Area of ΔABE = Area of ΔBDE

Similarly, in BEFC quadrilateral, BE||CF

With common base BC and between BE and CF parallel lines, we have

Area of ΔBEC = Area of ΔBEF

Adding both equations, we have

Area of ΔABE + Area of ΔBEC = Area of ΔBEF + Area of ΔBDE

=> Area of AEC = Area of DBF

Hence ProvedQuestion 17

In the given figure, ABCD is a parallelogram; BC is produced to point X. Prove that: area (Δ ABX) = area (quad. ACXD)

Solution 17

Given: ABCD is a parallelogram.

We know that

Area of ΔABC = Area of ΔACD

Consider ΔABX,

Area of ΔABX = Area of ΔABC + Area of ΔACX

We also know that area of triangles on the same base and between same parallel lines are equal.

Area of ΔACX = Area of ΔCXD

From above equations, we can conclude that

Area of ΔABX = Area of ΔABC + Area of ΔACX = Area of ΔACD+ Area of ΔCXD = Area of ACXD Quadrilateral

Hence ProvedQuestion 18

The given figure shows parallelograms ABCD and APQR. Show that these parallelograms are equal in area.

[Join B and R]

Solution 18

Join B and R and P and R.

We know that the area of the parallelogram is equal to twice the area of the triangle, if the triangle and the parallelogram are on the same base and between the parallels

Consider ABCD parallelogram:

Since the parallelogram ABCD and the triangle ABR lie on AB and between the parallels AB and DC, we have

....(1)

We know that the area of triangles with same base and between the same parallel lines are equal.

Since the triangles ABR and APR lie on the same base AR and between the parallels AR and QP, we have,

 ....(2)

From equations (1) and (2), we have,