ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area

Question 1.
Prove that the line segment joining the mid-points of a pair of opposite sides of a parallelogram divides it into two equal parallelograms.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q1.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q1.2

Question 2.
Prove that the diagonals of a parallelogram divide it into four triangles of equal area.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q2.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q2.2

Question 3.
(a) In the figure (1) given below, AD is median of ∆ABC and P is any point on AD. Prove that
(i) area of ∆PBD = area of ∆PDC.
(ii) area of ∆ABP = area of ∆ACP.
(b) In the figure (2) given below, DE || BC. prove that (i) area of ∆ACD = area of ∆ ABE.
(ii) area of ∆OBD = area of ∆OCE.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q3.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q3.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q3.3

Question 4.
(a) In the figure (1) given below, ABCD is a parallelogram and P is any point in BC. Prove that: Area of ∆ABP + area of ∆DPC = Area of ∆APD.
(b) In the figure (2) given below, O is any point inside a parallelogram ABCD. Prove that:
(i) area of ∆OAB + area of ∆OCD = \(\frac { 1 }{ 2 }\) area of || gm ABCD.
(ii) area of ∆ OBC + area of ∆ OAD = \(\frac { 1 }{ 2 }\) area of ||gmABCD
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q4.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q4.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q4.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q4.4

Question 5.
If E, F, G and H are mid-points of the sides AB, BC, CD and DA respectively of a parallelogram ABCD, prove that area of quad. EFGH = 1/2 area of || gm ABCD.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q5.1

Question 6.
(a) In the figure (1) given below, ABCD is a parallelogram. P, Q are any two points on the sides AB and BC respectively. Prove that, area of ∆ CPD = area of ∆ AQD.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q6.1
(b) In the figure (2) given below, PQRS and ABRS are parallelograms and X is any point on the side BR. Show that area of ∆ AXS = \(\frac { 1 }{ 2 }\) area of ||gm PQRS
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q6.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q6.3

Question 7.
D,EandF are mid-point of the sides BC, CA and AB respectively of a ∆ ABC. Prove that
(i) FDCE is a parallelogram
(ii) area of ADEF = \(\frac { 1 }{ 4 }\) area of A ABC
(iii) area of || gm FDCE = \(\frac { 1 }{ 2 }\) area of ∆ ABC.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q7.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q7.2

Question 8.
In the given figure, D, E and F are mid points of the sides BC, CA and AB respectively of AABC. Prove that BCEF is a trapezium and area of trap. BCEF = \(\frac { 3 }{ 4 }\) area of ∆ ABC.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q8.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q8.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q8.3

Question P.Q.
Prove that two triangles having equal areas and having one side of one of the triangles equal to one side of the other, have their corresponding altitudes equal.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp1.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp1.2

Question 9.
(a) In the figure (1) given below, the point D divides the side BC of ∆ABC in the ratio m : n. Prove that area of ∆ ABD: area of ∆ ADC = m : n.
(b) In the figure (2) given below, P is a point on the sidoBC of ∆ABC such that PC = 2BP, and Q is a point on AP such that QA = 5 PQ, find area of ∆AQC : area of ∆ABC.
(c) In the figure (3) given below, AD is a median of ∆ABC and P is a point in AC such that area of ∆ADP : area of AABD = 2:3. Find
(i) AP : PC (ii) area of ∆PDC : area of ∆ABC.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q9.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q9.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q9.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q9.4
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q9.5

Question 10.
(a) In the figure (1) given below, area of parallelogram ABCD is 29 cm2. Calculate the height of parallelogram ABEF if AB = 5.8 cm
(b) In the figure (2) given below, area of ∆ABD is 24 sq. units. If AB = 8 units, find the height of ABC.
(c) In the figure (3) given below, E and F are mid points of sides AB and CD respectively of parallelogram ABCD. If the area of parallelogram ABCip is 36 cm2.
(i) State the area of ∆ APD.
(ii) Name the parallelogram whose area is equal to the area of ∆ APD.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q10.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q10.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q10.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q10.4

Question 11.
(a) In the figure (1) given below, ABCD is a parallelogram. Points P and Q on BC trisect BC into three equal parts. Prove that :
area of ∆APQ = area of ∆DPQ = \(\frac { 1 }{ 6 }\) (area of ||gm ABCD)
(b) In the figure (2) given below, DE is drawn parallel to the diagonal AC of the quadrilateral ABCD to meet BC produced at the point E. Prove that area of quad. ABCD = area of ∆ABE.
(c) In the figure (3) given below, ABCD is a parallelogram. O is any point on the diagonal AC of the parallelogram. Show that the area of ∆AOB is equal to the area of ∆AOD.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q11.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q11.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q11.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q11.4
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q11.5

Question P.Q.
(a) In the figure (1) given below, two parallelograms ABCD and AEFB are drawn on opposite sides of AB, prove that: area of || gm ABCD + area of || gm AEFB = area of || gm EFCD.
(b) In the figure (2) given below, D is mid-point of the side AB of ∆ABC. P is any point on BC, CQ is drawn parallel to PD to meet AB in Q. Show that area of ∆BPQ = \(\frac { 1 }{ 2 }\) area of ∆ABC.
(c) In the figure (3) given below, DE is drawn parallel to the diagonal AC of the quadrilateral ABCD to meet BC produced at the point E. Prove that area of quad. ABCD = area of ∆ABE.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp2.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp2.2
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp2.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp2.4

Question 12.
(a) In the figure given, ABCD and AEFG are two parallelograms.
Prove that area of || gm ABCD = area of || gm AEFG.
(b) In the fig. (2) given below, the side AB of the parallelogram ABCD is produced to E. A st. line At through A is drawn parallel to CE to meet CB produced at F and parallelogram BFGE is Completed prove that area of || gm BFGE=Area of || gmABCD.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q12.1
(c) In the figure (3) given below AB || DC || EF, AD || BEandDE || AF. Prove the area ofDEFH is equal to the area of ABCD.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q12.2
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q12.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q12.4
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q12.5
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q12.6

Question 13.
Any point D is taken on the side BC of, a ∆ ABC and AD is produced to E such that AD=DE, prove that area of ∆ BCE = area of ∆ ABC.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q13.1

Question 14.
ABCD is a rectangle and P is mid-point of AB. DP is produced to meet CB at Q. Prove that area of rectangle ∆BCD = area of ∆ DQC.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q14.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q14.2

Question P.Q.
ABCD is a square, E and F are mid-points of the sides AB and AD respectively Prove that area of ∆CEF = \(\frac { 3 }{ 8 }\) (area of square ABCD).
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp3.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp3.1

Question P.Q.
A line PQ is drawn parallel to the side BC of ∆ABC. BE is drawn parallel to CA to meet QP (produced) at E and CF is drawn parallel to BA to meet PQ (produced) at F. Prove that
area of ∆ABE=area of ∆ACF.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp4.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp4.2

Question 15.
(a) In the figure (1) given below, the perimeter of parallelogram is 42 cm. Calculate the lengths of the sides of the parallelogram.
(b) In the figure (2) given below, the perimeter of ∆ ABC is 37 cm. If the lengths of the altitudes AM, BN and CL are 5x, 6x, and 4x respectively, Calculate the lengths of the sides of ∆ABC.
(c) In the fig. (3) given below, ABCD is a parallelogram. P is a point on DC such that area of ∆DAP = 25 cm² and area of ∆BCP = 15 cm². Find
(i) area of || gm ABCD
(ii) DP : PC.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q15.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q15.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q15.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q15.4
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q15.5

Question 16.
In the adjoining figure, E is mid-point of the side AB of a triangle ABC and EBCF is a parallelogram. If the area of ∆ ABC is 25 sq. units, find the area of || gm EBCF.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q16.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q16.2

Question 17.
(a) In the figure (1) given below, BC || AE and CD || BE. Prove that: area of ∆ABC= area of ∆EBD.
(b) In the llgure (2) given below, ABC is right angled triangle at A. AGFB is a square on the side AB and BCDE is a square on the hypotenuse BC. If AN ⊥ ED, prove that:
(i) ∆BCF ≅ ∆ ABE.
(ii)arca of square ABFG = area of rectangle BENM.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q17.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q17.2
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q17.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Q17.4

Multiple Choice Questions

Choose the correct answer from the given four options (1 to 8):
Question 1.
In the given figure, if l || m, AF || BE, FC ⊥ m and ED ⊥ m , then the correct statement is
(a) area of ||ABEF = area of rect. CDEF
(b) area of ||ABEF = area of quad. CBEF
(c) area of ||ABEF = 2 area of ∆ACF
(d) area of ||ABEF = 2 area of ∆EBD
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area mul Q1.1
Solution:
In the given figure,
l ||m, AF || BE, FC ⊥ m and ED ⊥ m
∵ ||gm ABEF and rectangle CDEF are on the same base EF and between the same parallel
∴ area ||gm ABEF = area rect. CDEF (a)

Question 2.
Two parallelograms are on equal bases and between the same parallels. The ratio of their areas is
(a) 1 : 2
(b) 1 : 1
(c) 2 : 1
(d) 3 : 1
Solution:
A triangle and a parallelogram are on the same base and between same parallel, then
∴ They are equal in area
∴ Their ratio 1:1 (b)

Question 3.
If a triangle and a parallelogram are on the same base and between same parallels, then the ratio of area of the triangle to the area of parallelogram is
(a) 1 : 3
(b) 1 : 2
(c) 3 : 1
(d) 1 : 4
Solution:
A triangle and a parallelogram are on the same base and between same parallel, then area of
triangle = \(\frac { 1 }{ 2 }\) area ||gm
∴ Their ratio 1 : 2 (b)

Question 4.
A median of a triangle divides it into two
(a) triangles of equal area
(b) congruent triangles
(c) right triangles
(d) isosceles triangles
Solution:
A median of a triangle divides it into two triangle equal in area. (a)

Question 5.
In the given figure, area of parallelogram ABCD is
(a) AB x BM
(b) BC x BN
(c) DC x DL
(d) AD x DL
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area mul Q5.1
Solution:
In the given figure,
Area of ||gm ABCD = AB x DL or DC x DL (∵ AB = DC) (c)

Question 6.
The mid-points of the sides of a triangle along with any of the vertices as the fourth point make a parallelogram of area equal to
(a) \(\frac { 1 }{ 2 }\) area of ∆ABC
(b) \(\frac { 1 }{ 3 }\) area of ∆ABC
(c) \(\frac { 1 }{ 4 }\) area of ∆ABC
(d) area of ∆ABC
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area mul Q6.1

Question 7.
In the given figure, ABCD is a trapezium with parallel sides AB = a cm and DC = b cm. E and F are mid-points of the non parallel sides. The ratio of area of ABEF and area of EFCD is
(a) a : b
(b) (3a + b) : (a + 3b)
(c) (a + 3b) : (3a + b)
(d) (2a + b) : (3a + b)
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area mul Q7.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area mul Q7.2

Question 8.
In the given figure, AB || DC and AB ≠ DC. If the diagonals AC and BD of the trapezium ABCD intersect at O, then which of the following statements is not true?
(a) area of ∆ABC = area of ∆ABD
(b) area of ∆ACD = area of ∆BCD
(c) area of ∆OAB = area of ∆OCD
(d) area of ∆OAD = area of ∆OBC
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area mul Q8.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area mul Q8.2

Chapter test

Question 1.
(a) In the figure (1) given below, ABCD is a rectangle (not drawn to scale ) with side AB = 4 cm and AD = 6 cm. Find :
(i) the area of parallelogram DEFC
(ii) area of ∆EFG.
(b) In the figure (2) given below, PQRS is a parallelogram formed by drawing lines parallel to the diagonals of a quadrilateral ABCD through its corners. Prove that area of || gm PQRS = 2 x area of quad. ABCD.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area ch Q1.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area ch Q1.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area ch Q1.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area ch Q1.4

Question P.Q.
In the adjoining figure, ABCD and ABEF are parallelogram and P is any point on DC. If area of || gm ABCD = 90 cm2, find:
(i) area of || gm ABEF
(ii) area of ∆ABP.
(iii) area of ∆BEF.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp5.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp5.2

Question 2.
In the parallelogram ABCD, P is a point on the side AB and Q is a point on the side BC. Prove that
(i) area of ∆CPD = area of ∆AQD
(ii)area of ∆ADQ = area of ∆APD + area of ∆CPB.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 2.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 2.2

Question 3.
In the adjoining figure, X and Y are points on the side LN of triangle LMN. Through X, a line is drawn parallel to LM to meet MN at Z. Prove that area of ∆LZY = area of quad. MZYX.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 3.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 3.2

Question P.Q.
If D is a point on the base BC of a triangle ABC such that 2BD = DC, prove that area of ∆ABD= \(\frac { 1 }{ 3 }\) area of ∆ ABC.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area Qp6.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area mul Qp6.2

Question 4.
Perpendiculars are drawn from a point within an equilateral triangle to the three sides. Prove that the sum of the three perpendiculars is equal to the altitude of the triangle.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 4.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 4.2

Question 5.
If each diagonal of a quadrilateral’ divides it into two triangles of equal areas, then prove that the quadrilateral is a parallelogram.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 5.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 5.2

Question 6.
In the given figure, ABCD is a parallelogram in which BC is produced to E such that CE = BC. AE intersects CD at F. If area of ∆DFB = 3 cm², find the area of parallelogram ABCD.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 6.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 6.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 6.3

Question 7.
In the given figure, ABCD is a square. E and F are mid-points of sides BC and CD respectively. If R is mid-point of EF, prove that: area of ∆AER = area of ∆AFR.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 7.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 7.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 7.3

Question 8.
In the given figure, X and Y are mid-points of the sides AC and AB respectively of ∆ABC. QP || BC and CYQ and BXP are straight lines. Prove that area of ∆ABP = area of ∆ACQ.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 8.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 14 Theorems on Area 8.2

 

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