M l aggarwal understanding icse mathematics class 9 Solutions for Chapter 14 theorems on area

Prove that the line segment joining the mid-points of a pair of opposite sides of a

parallelogram divides it into two equal parallelograms.

Prove that the diagonals of a parallelogram divide it into four triangles of equal

area.

(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.

(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 = ½ area of || gm ABCD

(ii) area of ∆ OBC + area of ∆ OAD = ½ area of || gm ABCD

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.

(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.

(b) In the figure (2) given below, PQRS and ABRS are parallelograms and X is any

the point on the side BR. Show that area of ∆ AXS = ½ area of ||gm PQRS.

D, E and F are mid-point of the sides BC, CA and AB respectively of a ∆ ABC.

Prove that

(i) FDCE is a parallelogram

(ii) area of ∆ DEF = ¼ area of ∆ ABC

(iii) area of || gm FDCE = ½ area of ∆ ABC

In the given figure, D, E, and F are midpoints of the sides BC, CA, and AB

respectively of ∆ ABC. Prove that BCEF is a trapezium and area of the trap. BCEF = ¾

area of ∆ ABC.

. (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 side BC 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 figure (3) given below, AD is a median of ∆ABC and P is a point in AC

such that area of ∆ADP: area of ∆ABD = 2:3. Find

(i) AP: PC

(ii) area of ∆PDC: area of ∆ABC.

(a) In the figure (1) given below, the area of parallelogram ABCD is 29 \mathrm{cm}^{2}

Calculate the height of parallelogram ABEF if AB = 5.8 cm

(b) In figure (2) given below, the area of ∆ABD is 24 sq. units. If AB = 8 units, find

the height of ABC.

(c) In figure (3) given below, E and F are midpoints of sides AB and CD

respectively of parallelogram ABCD. If the area of parallelogram ABC is 36 \mathrm{cm}^{2}

(i) State the area of ∆ APD.

(ii) Name the parallelogram whose area is equal to the area of ∆ APD.

(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 = 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 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.

. (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 straight line 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 || gm

ABCD.

(c) In the figure (3) given below AB || DC || EF, AD || BE and DE || AF. Prove the

area of DEFH is equal to the area of ABCD.

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.

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.

(a) In figure (1) given below, the perimeter of the 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.

In the adjoining figure, E is the 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.

(a) In the figure (1) given below, BC || AE and CD || BE. Prove that: area of

∆ABC= area of ∆EBD.

(b) In the figure (2) given below, ABC is a 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) area of square ABFG = area of rectangle BENM