Areas of Parallelograms and Triangles| NCERT Solutions| Class 9

NCERT Solutions of  Chapter: Areas of Parallelograms and Triangles. NCERT Solutions along with worksheets and notes for Class 9.

Exercise 9.1

(1) Which of the following figures lie on the same base and between the same parallels? In such a case, write the common base and the two parallels.

 

   

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(i)

Diagram of Question 1(i) of Exercise 9.1 of NCERT Chapter Areas of Parallelograms and Triangles

(ii)

Diagram of Question 1(ii) of Exercise 9.1 of NCERT Chapter Areas of Parallelograms and Triangles

Solution of question 1(ii) of NCERT chapter Areas of parallelograms and triangles

(iii)

Diagram of Question 1(iii) of Exercise 9.1 of NCERT Chapter Areas of Parallelograms and Triangles

Solution of question 1(iii) of exercise 9.1 of NCERT chapter Areas of parallelograms and triangles

(iv)

Diagram of Question 1(iv) of Exercise 9.1 of NCERT Chapter Areas of Parallelograms and Triangles

(v)

Diagram of Question 1(v) of Exercise 9.1 of NCERT Chapter Areas of Parallelograms and Triangles

(vi)

Diagram of Question 1(vi) of Exercise 9.1 of NCERT Chapter Areas of Parallelograms and Triangles

 

Diagram of Question 1 of Exercise 9.2 of NCERT Chapter Areas of Parallelograms and Triangles

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Diagram of Question 2 of Exercise 9.2 of NCERT Chapter Areas of Parallelograms and Triangles

(3)P and Q are any two points lying on the sides DC and AD respectively of a parallelogram ABCD. Show that ar(APB) = ar(BQC).

 

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Diagram of Question 3 of Exercise 9.2 of NCERT Chapter Areas of Parallelograms and Triangles

Diagram of Question 4 of Exercise 9.2 of NCERT Chapter Areas of Parallelograms and Triangles

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(i)

(ii)

Diagram of Question 5 of Exercise 9.2 of NCERT Chapter Areas of Parallelograms and Triangles

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(6) A farmer was having a field in the form of a parallelogram PQRS. She took any point A on RS and joined it to points P and Q. In how many parts the fields is divided? What are the shapes of these parts? The farmer wants to sow wheat and pulses in equal portions of the field separately. How should she do it?

 

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Diagram of Question 6 of Exercise 9.2 of NCERT Chapter Areas of Parallelograms and Triangles

Diagram of Question 1 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

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Diagram of Question 2 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

(3) Show that the diagonals of a parallelogram divide it into four triangles of equal area.

 

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Diagram of Question 3 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

(4) In figure below, 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).

 

Diagram of Question 4 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

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Diagram of Question 5 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

Diagram of Question 6 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

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Draw perpendiculars DE and BF from points D and B to AC respectively.

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Diagram of Question 7 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

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Diagram of Question 8 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

(9) 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 (Figure below). Show that ar(ABCD) = ar (PBQR).

[Hint: Join AC and PQ. Now compare ar(ACQ) and ar(APQ).]

 

Diagram of Question 9 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

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Diagram of Question 10 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

Diagram of Question 11 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

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(12)A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decided to rake 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 equal amount of land in lieu of his land adjoining his plot so as to from a triangular plot. Explain how this proposal will be implemented.

 

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Diagram of Question 12 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

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Diagram of Question 13 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

Diagram of Question 14 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

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

 

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Diagram of Question 15 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

(16) In figure below, ar(DRC) = ar(DPC) and ar(BDP) = ar(ARC). Show that both the quadrilaterals ABCD and DCPR are trapeziums.

 

Diagram of Question 16 of Exercise 9.3 of NCERT Chapter Areas of Parallelograms and Triangles

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Exercise 9.4

(1) 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.

 

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Diagram of Question 1 of Exercise 9.4 of NCERT Chapter Areas of Parallelograms and Triangles

(2) In figure below, D and E are two points on BC such that BD = DE = EC. Show that ar(ABD) = ar(ADE) = ar(AEC).

Can you now answer the question that you have left in the ‘Introduction’ of this chapter, whether the field of the Budhia has been actually divided into three parts of equal area?

[Remark: Note that by taking BD = DE = EC, the triangle ABC is divided into three triangles ABD, ADE and AEC of equal areas. In the same way, by dividing BC into n equal parts and joining the points of division so obtained to the opposite vertex of BC, you can divide ABC into n triangles of equal areas.]

 

Diagram of Question 2 of Exercise 9.4 of NCERT Chapter Areas of Parallelograms and Triangles

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(3)In figure below, ABCD and DCFE and ABFE are parallelograms. Show that ar(ADE) = ar(BCF).

Diagram of Question 3 of Exercise 9.4 of NCERT Chapter Areas of Parallelograms and Triangles

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(4) In figure below, ABCD is a parallelogram and BC is produced to a point Q such that AD = CQ. If AQ intersects DC at P, show that ar(BPC) = ar(DPQ).

[Hint: Join AC]

 

Diagram of Question 4 of Exercise 9.4 of NCERT Chapter Areas of Parallelograms and Triangles

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Diagram of Question 5 of Exercise 9.4 of NCERT Chapter Areas of Parallelograms and Triangles

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(i)

(ii)

(iii)

(v)

(vi)

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Diagram of Question 6 of Exercise 9.4 of NCERT Chapter Areas of Parallelograms and Triangles

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Diagram of Question 7 of Exercise 9.4 of NCERT Chapter Areas of Parallelograms and Triangles

(i)

(ii)

 

Diagram of Question 8 of Exercise 9.4 of NCERT Chapter Areas of Parallelograms and Triangles

 

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(ii)

(iii)

(iv)

(v)

(vi)

Helping Topics

Areas of Parallelograms and Triangles

Worksheet Class 9

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