#### Chapters

Chapter 2: Exponents of Real Numbers

Chapter 3: Rationalisation

Chapter 4: Algebraic Identities

Chapter 5: Factorisation of Algebraic Expressions

Chapter 6: Factorisation of Polynomials

Chapter 7: Introduction to Euclidβs Geometry

Chapter 8: Lines and Angles

Chapter 9: Triangle and its Angles

Chapter 10: Congruent Triangles

Chapter 11: Co-ordinate Geometry

Chapter 12: Heronβs Formula

Chapter 13: Linear Equations in Two Variables

Chapter 14: Quadrilaterals

Chapter 15: Areas of Parallelograms and Triangles

Chapter 16: Circles

Chapter 17: Constructions

Chapter 18: Surface Areas and Volume of a Cuboid and Cube

Chapter 19: Surface Areas and Volume of a Circular Cylinder

Chapter 20: Surface Areas and Volume of A Right Circular Cone

Chapter 21: Surface Areas and Volume of a Sphere

Chapter 22: Tabular Representation of Statistical Data

Chapter 23: Graphical Representation of Statistical Data

Chapter 24: Measures of Central Tendency

Chapter 25: Probability

#### RD Sharma Mathematics Class 9 by R D Sharma (2018-19 Session)

## Chapter 15 : Areas of Parallelograms and Triangles

#### Page 0

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.

#### Page 0

In fig below, ABCD is a parallelogram, AE ⊥ DC and CF ⊥ AD. If AB = 16 cm, AE = 8

cm and CF = 10 cm, find AD.

In Q. No 1, if AD = 6 cm, CF = 10 cm, and AE = 8cm, find AB.

Let ABCD be a parallelogram of area 124 cm2. If E and F are the mid-points of sides AB and

CD respectively, then find the area of parallelogram AEFD.

If ABCD is a parallelogram, then prove that

ππ (Δπ΄π΅π·) = ππ (Δπ΅πΆπ·) = ππ (Δπ΄π΅πΆ) = ππ (Δπ΄πΆπ·) = `1/2` ππ (||^{ππ} π΄π΅πΆπ·) .

#### Page 0

In the below figure, compute the area of quadrilateral ABCD.

In the below figure, PQRS is a square and T and U are respectively, the mid-points of PS

and QR. Find the area of ΔOTS if PQ = 8 cm.

Compute the area of trapezium PQRS is Fig. below.

In the below fig. ∠AOB = 90°, AC = BC, OA = 12 cm and OC = 6.5 cm. Find the area of

ΔAOB.

In the below fig. ABCD is a trapezium in which AB = 7 cm, AD = BC = 5 cm, DC = x cm,

and distance between AB and DC is 4cm. Find the value of x and area of trapezium ABCD.

In the below fig. OCDE is a rectangle inscribed in a quadrant of a circle of radius 10 cm. If

OE = 2√5, find the area of the rectangle.

In the below fig. ABCD is a trapezium in which AB || DC. Prove that ar (ΔAOD) =

ar(ΔBOC).

In the given below fig. ABCD, ABFE and CDEF are parallelograms. Prove that ar (ΔADE)

= ar (ΔBCF)

Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that:

ar(ΔAPB) × ar(ΔCPD) = ar(ΔAPD) × ar (ΔBPC)

In the below Fig, ABC and ABD are two triangles on the base AB. If line segment CD is

bisected by AB at O, show that ar (Δ ABC) = ar (Δ ABD)

If P is any point in the interior of a parallelogram ABCD, then prove that area of the

triangle APB is less than half the area of parallelogram.

If AD is a median of a triangle ABC, then prove that triangles ADB and ADC are equal in

area. If G is the mid-point of median AD, prove that ar (Δ BGC) = 2 ar (Δ AGC).

A point D is taken on the side BC of a ΔABC such that BD = 2DC. Prove that ar(Δ ABD) =

2ar (ΔADC).

ABCD is a parallelogram whose diagonals intersect at O. If P is any point on BO, prove

that: (1) ar (ΔADO) = ar (ΔCDO) (2) ar (ΔABP) = ar (ΔCBP)

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

CD at F.

(i) Prove that ar (ΔADF) = ar (ΔECF)

(ii) If the area of ΔDFB = 3 cm2, find the area of ||^{gm} ABCD.

ABCD is a parallelogram whose diagonals AC and BD intersect at O. A line through O

intersects AB at P and DC at Q. Prove that ar (Δ POA) = ar (Δ QOC).

ABCD is a parallelogram. E is a point on BA such that BE = 2 EA and F is a point on DC

such that DF = 2 FC. Prove that AE CF is a parallelogram whose area is one third of the

area of parallelogram ABCD.

In a ΔABC, P and Q are respectively the mid-points of AB and BC and R is the mid-point

of AP. Prove that :

(1) ar (Δ PBQ) = ar (Δ ARC)

(2) ar (Δ PRQ) =`1/2`ar (Δ ARC)

(3) ar (Δ RQC) =`3/8` ar (Δ ABC) .

ABCD is a parallelogram, G is the point on AB such that AG = 2 GB, E is a point of DC

such that CE = 2DE and F is the point of BC such that BF = 2FC. Prove that:

(1) ar ( ADEG) = ar (GBCD)

(2) ar (ΔEGB) = `1/6` ar (ABCD)

(3) ar (ΔEFC) = `1/2` ar (ΔEBF)

(4) ar (ΔEBG) = ar (ΔEFC)

^{(5)ΔFind} what portion of the area of parallelogram is the area of οEFG.

In Fig. below, CD || AE and CY || BA.

(i) Name a triangle equal in area of ΔCBX

(ii) Prove that ar (Δ ZDE) = ar (ΔCZA)

(iii) Prove that ar (BCZY) = ar (Δ EDZ)

In below fig., PSDA is a parallelogram in which PQ = QR = RS and AP || BQ || CR. Prove

that ar (Δ PQE) = ar (ΔCFD).

In the below fig. ABCD is a trapezium in which AB || DC and DC = 40 cm and AB = 60

cm. If X and Y are respectively, the mid-points of AD and BC, prove that:

(i) XY = 50 cm

(ii) DCYX is a trapezium

(iii) ar (trap. DCYX) =`9/11`ar (trap. (XYBA))

In Fig. below, ABC and BDE are two equilateral triangles such that D is the mid-point of

BC. AE intersects BC in F. Prove that

(1) ar (Δ BDE) = `1/2` ar (ΔABC)

(2) Area ( ΔBDE) `= 1/2 ` ar (ΔBAE)

(3) ar (BEF) = ar (ΔAFD)

(4) area (Δ ABC) = 2 area (ΔBEC)

(5) ar (ΔFED) `= 1/8` ar (ΔAFC)

(6) ar (Δ BFE) = 2 ar (ΔEFD)

D is the mid-point of side BC of ΔABC and E is the mid-point of BD. if O is the mid-point

of AE, prove that ar (ΔBOE) = `1/8` ar (Δ ABC).

In the below fig. X and Y are the mid-points of AC and AB respectively, QP || BC and

CYQ and BXP are straight lines. Prove that ar (Δ ABP) = ar (ΔACQ).

In the below fig. ABCD and AEFD are two parallelograms. Prove that

(1) PE = FQ

(2) ar (Δ APE) : ar (ΔPFA) = ar Δ(QFD) : ar (Δ PFD)

(3) ar (ΔPEA) = ar (ΔQFD)

In the below figure, ABCD is parallelogram. O is any point on AC. PQ || AB and LM ||

AD. Prove that ar (||gm DLOP) = ar (||^{gm} BMOQ)

In a ΔABC, if L and M are points on AB and AC respectively such that LM || BC. Prove

that:

(1) ar (ΔLCM ) = ar (ΔLBM )

(2) ar (ΔLBC) = ar (ΔMBC)

(3) ar (ΔABM) ar (ΔACL)

(4) ar (ΔLOB) ar (ΔMOC)

In the below fig. D and E are two points on BC such that BD = DE = EC. Show that ar

(ΔABD) = ar (ΔADE) = ar (ΔAEC).

In the following figure, 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 AX ⊥ DE meets BC at Y. Show that:-

(i) ΔMBC ≅ ΔABD

(ii) ar (BYXD) = 2 ar(MBC)

(iii) ar (BYXD) = ar(ABMN)

(iv) ΔFCB ≅ ΔACE

(v) ar(CYXE) = 2 ar(FCB)

(vi) ar (CYXE) = ar(ACFG)

(vii) ar (BCED) = ar(ABMN) + ar(ACFG)

Note : Result (vii) is the famous Theorem of Pythagoras. You shall learn a simpler proof of this theorem in Class X.

#### RD Sharma Mathematics Class 9 by R D Sharma (2018-19 Session)

#### Textbook solutions for Class 9

## RD Sharma solutions for Class 9 Mathematics chapter 15 - Areas of Parallelograms and Triangles

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Concepts covered in Class 9 Mathematics chapter 15 Areas of Parallelograms and Triangles are Triangles on the Same Base and Between the Same Parallels, Parallelograms on the Same Base and Between the Same Parallels, Figures on the Same Base and Between the Same Parallels, Introduction to Areas of Parallelograms and Triangles.

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