# Solution of Chapter 4. Triangles (RD Sharma - Mathematics Book)

## Exercise 4.1

1

Fill in the blanks using the correct word given in brackets :

(i) All circles are……..(congruent, similar).

(ii) All squares are………(similar, congruent).

(iii) All……triangles are similar (isosceles, equilaterals).

(iv) Two triangles are similar, if heir corresponding angles are………(proportional, equal)

(v) Two triangles are similar, if their corresponding sides are………(proportional, equal)

(vi) Two polygons of the same number of sides are similar, if (a) their corresponding angles ae and (b) heir corresponding sides are………(equal, proportional)

2

Write the truth value (T/F) of each of the following statements:

(i) Any two similar figures are congruent.

(ii) Any two congruent figures are similar.

(iii) Two polygons are similar, if their corresponding sides are proportional.

(iv) Two polygons are similar if their corresponding angles are proportional.

(v) Two triangles are similar if their corresponding sides are proportional.

(vi) Two triangles are similar if their corresponding angles are proportional.

## Exercise 4.2

1

In a , D and E are points on the sides AB and AC respectively such that

(i) If AD = 6 cm, DB = 9 cm and AE = 8 cm, find AC.

(ii) If and AC = 15 cm, find AE.

(iii) If and AC = 18 cm, find AE.

(iv) If AD = 4, AE = 8, DB = x – 4, and EC = 3x – 19, find x.

(v) If AD = 8 cm, AB = 12 cm and AE = 12 cm, find CE.

(vi) If AD = 4 cm, DB = 4.5 cm and AE = 8 cm, find AC.

(vii) If AD = 2 cm, AB = 6 cm and AC = 9 cm, find AE.

(viii) If and EC = 2.5 cm, find AE.

(ix) If AD = x, DB = x – 2, AE = x + 2 and EC = x – 1, find the value of x.

(x) If AD = 8x - 7, DB = 5x – 3, AE = 4x - 3 and EC = (3x – 1), find the value of x.

(xi) If AD = 4x – 3, AE = 8x – 7, BD = 3x – 1 and CE = 5x - 3, find the volume x.

(xii) If AD = 2.5 cm, BD = 3.0 cm and AE = 3.75 cm, find the length of AC.

2

In a, D and E are points on the sides AB and AC respectively. For each of the following cases show that :

(i) AB = 12 cm, AD = 8 cm, AE = 12 cm and AC = 18 cm.

(ii) AB = 5.6 cm, AD = 1.4 cm, AE = 7.2 cm and AC = 1.8 cm.

(iii) AB = 10.8 cm, BD = 4.5 cm, AC = 4.8 cm and AE = 2.8 cm.

(iv) AD = 5.7 cm, BD = 9.5 cm, AE = 3.3 cm and EC = 5.5 cm.

3

In a, P and Q are points on sides AB and AC respectively, such that . If AP = 2.4 cm, AQ = 2 cm, QC = 3 cm and BC = 6 cm, find AB and PQ.

4

In a ΔABC, D and E are points on AB and AC respectively such that DE||BC. If AD = 2.4 cm, AE = 3.2 cm, DE = 2 cm and BC = 5 cm, find BD and CE.

5

In Fig. 4.35, state if .

6

M and N are points on the sides PQ and PR respectively of a . For each of the following cases, state whether :

(i) PM = 4 cm, QM = 4.5 cm, PN = 4 cm, NR = 4.5 cm

(ii) PQ = 1.28 cm, PR = 2.56 cm, PM = 0.16 cm, PN = 0.32 cm

7

In three line segments OA, OB, and OC, points L, M, N respectively are so chosen that and but neither of L, M, N nor of A, B, C are collinear. Show that .

8

If D and E are points on sides AB and AC respectively of a such that and BD = CE. Prove that is isosceles.

## Exercise 4.3

1

In a , AD is the bisector of , meeting side BC at D.

(i) If BD = 2.5 cm, AB = 5 cm and AV = 4.2 cm, find DC.

(ii) If BD = 2 cm, AB = 5 cm and DC = 3 cm, find AC.

(iii) If AB = 3.5 cm, AC = 4.2 cm and DC = 2.8 cm, find BD.

(iv) If AB = 10 cm, AC = 14 cm and BC = 6 cm, find BD and DC.

(v) If AC = 4.2 cm, DC = 6 cm and BC = 10 cm, find AB.

(vi) If AB = 5.6 cm, AC = 6 cm and DC = 6 cm, find BC.

(vii) If AD = 5.6 cm, BC = 6 cm and BD = 3.2 cm, find AC.

(viii) If AB = 10 cm, AC = 6 cm and BC = 12 cm, find BD and DC.

2

In Fig. 4.57, AE is the bisector of the exterior meeting BC produced in E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, find CE.

3

In Fig. 4.58, is a triangle such that , . Find.

4

In (fig. 4.59), if , prove that .

5

D, E and F are the points on sides BC, CA and AB respectively of such that AD bisects , BE bisects and CF bisects . If AB = 5 cm, BC = 8 cm and CA = 4 cm, determine AF, CE and BD.

6

In Fig. 4.60, check whether AD is the bisector of of in each of the following:

(i) AB = 5 cm, AC = 10 cm, BD = 1.5 cm and CD = 3.5 cm

(ii) AB = 4 cm, AC = 6 cm, BD = 1.6 cm and CD = 2.4 cm

(iii) AB = 8 cm, AC = 24 cm, BD = 6 cm and BC = 24 cm

(iv) AB = 6 cm, AC = 8 cm, BD = 1.5 cm and CD = 2 cm

(v) AB = 5 cm, AC = 12 cm, BD = 2.5 cm and BC = 9 cm

7

In Fig. 4.60, AD bisects , AB = 12 cm, AC = 20 cm and BD = 5 cm, determine CD.

## Exercise 4.4

1

(i) In fig. 4.70, if , find the value of x.

1

In Fig. 4.71, if , find the value of x.

1

In Fig. 4.72, . If OA = 3x – 19, OB = x – 4, OC = x – 3 and OD = 4, find x.

## Exercise 4.5

1

In Fig. 4.136, . If BC = 8 cm, PQ = 4 cm, BA = 6.5 cm and AP = 2.8 cm, find CA and AQ

2

A vertical stick 10 cm long casts a shadow 8 cm long. At the same time a tower casts a shadow 30 m long. Determine the height of the tower.

3

In Fig. 4.137, . Find the length of PB.

4

In Fig. 4.138, . Find the length of XY.

5

In a right angled triangle with sides a and b and hypotenuse c, the altitude drawn on the hypotenuse is x. Prove that AB = CX.

6

In Fig. 4.139, = 90° and . If BD = 8 cm and AD = 4 cm, find CD.

7

In Fig. 4.140, = 90° and . If AB = 5.7 cm, BD = 3.8 cm and CD = 5.4 cm, find BC.

8

In Fig. 4.141 such that AE = (1/4) AC. If AB = 6 cm, find AD.

9

In Fig. 4.142, PA, QB and RC are each perpendicular to AC. Prove that .

10

In Fig. 4.143, , prove that . Also, find the value of x.

11

The perimeters of two similar triangles are 25 cm and 15 cm respectively. If one side of first triangle is 9 cm, what is the corresponding side of the other triangle?

12

In , it is being given that: AB = 5 cm, BC = 4 cm and CA = 4.2 cm; DE = 10 cm, EF = 8 cm and FD = 8.4 cm. If and , find AL : DM.

13

D and E are the points on the sides AB and AC respectively of a such that AD = 8 cm, DB = 12 cm, AE = 6 cm and CE = 9 cm. Prove that BC = 5/2 DE.

14

D is the mid-point of side BC of a . AD is bisected at the point E and BE produced cuts AC at the point X. Prove that BE : EX = 3 : 1.

15

ABCD is a parallelogram and APQ is a straight line meeting BC at P and DC produced at Q. Prove that the rectangle obtained by BP and DQ is equal to the rectangle contained by AB and BC.

16

In , AL and CM are the perpendiculars from the vertices A and C to BC and AB respectively. If AL and CM intersect at O, prove that :

(i)

(ii)

17

In fig. 4.144, we have  AB||CD||EF. If AB = 6 cm, CD = x cm, EF = 10 cm, BD = 4 cm and DE = y cm, calculate the values of x and y.

18

ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the mid-points of AB, AC, CD and BD respectively, show that PQRS is a rhombus.

19

In Fig. 4.145, If and , prove that .

20

In an isosceles , the base AB is produced both the ways to P and Q such that AP × BQ = AC2. Prove that .

21

A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/sec. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds.

22

Diagonals AC and BD of a trapezium ABCD with intersect each other at the point O. Using similarity criterion for two triangles, show that .

23

If and are two right triangles, right angled at B and M respectively such that . Prove that

(i)

(ii)

24

A vertical stick of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.

25

In Fig. 4.145 (a) is right angled at C and . Prove that and hence find the lengths of AE and DE.

## Exercise 4.6

1

Triangles ABC and DEF are similar.

(i) If area () = 16 cm2, area () = 25 cm2 and BC = 2.3 cm, find EF.

(ii) If area () = 9 cm2, area () = 64 cm2 and DE = 5.1 cm, find AB.

(iii) If AC = 19 cm and DF = 8 cm, find the ratio of the area of two triangles.

(iv) If area () = 36 cm2, area () = 64 cm2 and DE = 6.2 cm, find AB.

(v) If AB = 1.2 cm and DE = 1.4 cm, find the ratio of the areas of .

2

In Fig. 4.177, . If BC = 10 cm, PQ = 5 cm, BA = 6.5 cm and AP = 2.8 cm, find CA and AQ. Also, find the area () : area ().

3

The areas of two similar triangles are 81 cm2 and 49 cm2 respectively. Find the ratio of their corresponding heights. What is the ratio of their corresponding medians?

4

The areas of two similar triangles are 169 cm2 and 121 cm2 respectively. If the longest side of the larger triangle is 26 cm, find the longest side of the smaller triangle.

5

Two isosceles triangles have equal vertical angles and their areas are in the ratio 36 : 25.. Find the ratio of their corresponding heights.

6

The areas of two similar triangles are 25 cm2 and 36 cm2 respectively. If the altitude of the first triangle is 2.4 cm, find the corresponding altitude of the other.

7

The corresponding altitudes of two similar triangles are 6 cm and 9 cm respectively. Find the ratio of their areas.

8

ABC is a triangle in which = 90°, , BC = 12 cm and AC = 5 cm. Find the ratio of the areas of .

9

In Fig. 4.178,

(i) If DE = 4 cm, BC = 6 cm and area () = 16 cm2, find the area of .

(ii) If DE = 4 cm, BC = 8 cm and area () = 25 cm2, find the area of .

(iii) If DE : BC = 3 : 5. Calculate the ratio of the areas of and the trapezium BCED.

10

In , D and E are the mid-points of AB and AC respectively. Find the ratio of the areas of .

11

In Fig. 4.179, are on the same base BC. If AD and BC intersect at O. Prove that

12

ABCD is a trapezium in which . The diagonals AC and BD intersect at O. Prove that : (i)

(ii) If OA = 6 cm, OC = 8 cm, Find:

(a) (b)

13

In , P divides the side AB such that AP : PB = 1 : 2. Q is a point in AC such that . Find the ratio of the areas of and trapezium BPQC.

14

The areas of two similar triangles are 100 cm2 and 49 cm2 respectively. If the altitude of the bigger triangle is 5 cm, find the corresponding altitude of the other.

15

The areas of two similar triangles are 121 cm2 and 64 cm2 respectively. If the median of the first triangle is 12.1 cm, find the corresponding median of the other.

16

If such that AB = 5 cm, area () = 20 cm2 and area () = 45 cm2 , determine DE.

17

In , PQ is a line segment intersecting AB at P and AC at Q such that and PQ divides into two parts equal in area. Find .

18

The areas of two similar triangles ABC and PQR are in the ratio 9 : 16. If BC = 4.5 cm, find the length of QR.

19

ABC is a triangle and PQ is a straight line meeting AB in P and AC in Q. If AP = 1 cm, PB = 3 cm, AQ = 1.5 cm, QC = 4.5 m, prove that area of is one-sixteenth of the area of .

20

If D is a point on the side AB of such that AD : DB = 3.2 and E is a point on BC such that . Find the ratio of areas of .

21

If are equilateral triangles, where D is the mid point of BC, find the ratio of areas of .

22

AD is and altitude of an equilateral triangle ABC. On AD as base, another equilateral triangle ADE is constructed. Prove that Area () : Area () = 3 : 4.

## Exercise 4.7

1

If the sides of a triangle are 3 cm, 4 cm and 6 cm long, determine whether the triangle is a right-angled triangle.

2

The sides of certain triangles are given below. Determine which of them are right triangles.

(i) a = 7 cm, b = 24 cm and c = 25 cm

(ii) a = 9 cm, b = 16 cm and c = 18 cm

(iii) a = 1.6 cm, b = 3.8 cm and c = 4 cm

(iv) a = 8 cm, b = 10 cm and c = 6 cm

3

A man goes 15 metres due west and then 8 metres due north. How far is he from the starting point?

4

A ladder 17 m long reaches a window of a building 15 m above the ground. Find the distance of the foot of the ladder from the building.

5

Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.

6

In an isosceles triangle ABC, AB = AC = 25 cm, BC = 14 cm. Calculate the altitude from A on BC.

7

The foot of a ladder is 6 m away from a wall and its top reaches a window 8 m above the ground. If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its tip reach?

8

Two poles of height 9 m and 14 m stand on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.

9

Using Pythagoras theorem determine the length of AD in terms of b and c shown in Fig 4.219.

10

A triangle has sides 5 cm, 12 cm and 13 cm. Find the length to one decimal place, of the perpendicular from the opposite vertex to the side whose length is 13 cm.

11

ABCD is a square. F is the mid-point of AB. BE is one third of BC. If the area of ΔFBE = 108 cm2, find the length of AC.

12

In an isosceles triangle ABC, if AB = AC = 13 cm and the altitude from A on BC is 5 cm, find BC.

13

In a , AB = BC = CA = 2 a and . Prove that

(i) (ii) Area () =

14

The lengths of the diagonals of a rhombus are 24 cm and 10 cm. Find each side of the rhombus.

15

Each side of a rhombus is 10 cm. If one of its diagonals is 16 cm find the length of the other diagonal.

16

In an acute-angled triangle, express a median in terms of its sides.

17

Calculate the height of an equilateral triangle each of whose sides measures 12 cm.

18

In right-angled triangle ABC in which, if D is the mid-point of BC, prove that .

19

In Fig. 4.220, D is the mid-point of side BC and . If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that:

(i) (ii)

(iii)

20

In Fig. 4.221, and segment , show that

(i)

(ii)

21

In , is obtuse, and . Prove that:

(i)

(ii)

22

In a right right-angled at C, if D is the mid-point of BC, prove that .

23

In a quadrilateral ABCD, , , prove that .

24

In an equilateral, , prove that .

25

is a right triangle right-angled at A and . Show that

(i) (ii)

(iii) (iv)

26

A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?

27

An aeroplane leaves an airport and files due north at a speed of 1000 km/hr. At the same time, another aeroplane leaves the same airport and files due west at a speed of 1200 km/hr. How far apart will be the two planes after hours?

28

Determine whether the triangle having sides (a – 1) cm, cm and (a + 1) cm is a right angled triangle.

## CCE - Formative Assessment

1

State basic proportionality theorem and its converse.

2

In the adjoining figure, find AC.

3

In the adjoining figure, if AD is the bisector of ∠A, what is AC?

4

State AAA similarity criterion.

5

State SSS similarity criterion.

6

State SAS similarity criterion.

7

In the adjoining figure, DE is parallel to BC and AD = 1 cm, BD = 2 cm. What is the ratio of the area of A ABC to the area of A ADE?

8

In the figure given below. If AD = 2.4 cm, DB = 3.6 cm and AC = 5 cm. Find AE.

9

If the areas of two similar triangles ABC and PQR are in the ratio 9 : 16 and BC = 4.5 cm, what is the length of QR?

10

The areas of two similar triangles are 169 cm2 and 121 cm2 respectively. If the longest side of the larger triangle is 26 cm, what is the length of the longest side of the smaller triangle?

11

If ABC and DEF are similar triangles such that = 57° and = 73°, what is the measure of ?

12

If the altitude of two similar triangles are in the ratio 2 : 3, what is the ratio of their areas?

13

If and are two triangles such that , then write Area (): Area ().

14

If and are similar triangles such that AB = 3 cm, BC = 2 cm CA = 2.5 cm and EF = 4 cm, write the perimeter of.

15

State Pythagoras theorem and its converse.

16

The lengths of the diagonals of a rhombus are 30 cm and 40 cm. Find the side of the rhombus. [CBSE 2008]

17

In Fig. 4.236, and AP : PB = 1 : 2. Find [CBSE 2008]

18

In Fig. 4.237, LM = LN = 46°. Express x in terms of a, b and c where a, b, c are lengths of LM, MN and and NK respectively.

19

In Fig. 4.238, S and T are points on the sides PQ and PR respectively of A PQR such that PT = 2 cm, TR = 4 cm and ST is parallel to QR. Find the ratio of the areas of and . [CBSE 2010]

20

In Fig. 4.239, is similar to. If AK = 10 cm, BC = 3.5 cm and HK = 7 cm, find AC. [CBSE 2010]

21

In Fig. 4.240, in such that BC = 8 cm, AB = 6 cm and DA =1.5 cm. Find DE.

22

In Fig. 4.241, and AD = BD. If BC = 4.5 cm, find DE.

1

A vertical stick 20 m long casts a shadow 10 m long on the ground. At the same time, a tower casts a shadow 50 m long on the ground. The height of the tower is

2

Sides of two similar triangles are in the ratio 4 : 9 . Areas of these triangles are in the ratio.

3

The areas of two similar triangles are in respectively 9 cm2 and 16 cm2. The ratio of their corresponding sides is

4

The areas of two similar triangles and are 144 cm2 and 81 cm2 respectively. If the longest side of larger A ABC be 36 cm, then. the longest side of the smaller triangle is

5

and are two equilateral triangles such that D is the mid-point of BC. The ratio of the areas of triangles ABC and BDE is

6

Two isosceles triangles have equal angles and their areas are in the ratio 16 : 25. The ratio of their corresponding heights is

7

If and are similar such that 2 AB = DE and BC = 8 cm, then EF =

8

If and are two triangles such that , then Area (): Area () =

9

is such that AB = 3 cm, BC = 2 cm and CA = 2 . 5 cm. If and EF = 4 cm, then perimeter of is

10

XY is drawn parallel to the base BC of cutting AB at X and AC at Y. If AB = 4 BX and YC = 2 cm, then AY =

11

Two poles of height 6 m and 11 m stand vertically upright on a plane ground. If the distance between their foot is 12 m, the distance between their tops is

12

In , a line XY parallel to BC cuts AB at X and AC at Y. If BY bisects, then

A. BC =CY

B. BC = BY

C. BC CY

D. BC BY

13

In , D and E are points on side AB and AC respectively such that and AD: DB = 3 : 1. If EA = 3.3 cm, then AC =

14

In triangles ABC and DEF, AB : ED = AC : EF and = 65°, then =

15

If ABC and DEF are similar triangles such that = 47° and = 83°, then =

16

If D, E, F are the mid-points of sides BC, CA and AB respectively of A ABC, then the ratio of the areas of triangles DEF and ABC is

17

In a , = 90°, AB = 5 cm and AC = 12 cm. If , then AD =

18

In an equilateral triangle ABC, if, then

19

If is an equilateral triangle such that, then AD2 =

20

In a , perpendicular AD from A on BC meets BC at D. If BD = 8 cm, DC = 2 cm and AD = 4 cm, then

21

In a , point D is on side AB and point E is on side AC, such that BCED is a trapezium. If DE : BC = 3 : 5, then Area (): Area (âBCED) =

22

In a , AD is the bisector of . If AB = 6 cm, AC = 5 cm and BD = 3 cm then DC =

23

In a , AD is the bisector of. If AB = 8 cm, BD = 6 cm and DC = 3 cm. Find AC

24

ABCD is a trapezium such that and AB = 4 cm. If the diagonals AC and BD intersect at O such that , then BC =

25

If ABC is an isosceles triangle and D is a point on BC such that, then

26

is a right triangle right-angled at A and . Then, =

27

If ABC is a right triangle right-angled at B and M, N are the mid-points of AB and BC respectively, then 4 (AN2 + CM2) =

28

If E is a point on side CA of an equilateral triangle ABC such that, then AB2 + BC + CA2 =

29

In a right triangle ABC right-angled at B, if P and Q are points on the sides AB and AC respectively, then

30

If in and , , then when

31

If in two triangles ABC and DEF, , then

32

, ar () = 9 cm2, ar () = 16 cm2. If BC = 2.1 cm, then the measure of EF is

33

The length of the hypotenuse of an isosceles right triangle whose one side is 4 cm is

34

A man goes 24 m due west and then 7 m due north. How far is he from the starting point?

35

. If BC = 3 cm, EF = 4 cm and ar () = 54 cm2, then ar () =

36

. such that ar () = 4 ar (). If BC =12 cm, then QR =

37

The areas of two similar triangles are 121 cm2 and 64 cm2 respectively. If the median of the first triangle is 12.1 cm, then the corresponding median of the other triangle is

38

If such that DE = 3 cm, EF = 2 cm, DF = 2.5 cm, BC = 4 cm, then perimeter of is

39

In an equilateral triangle ABC if , then AD2 =

40

In an equilateral triangle ABC if , then

41

If such that AB = 9.1 cm and DE = 6.5 cm. If the perimeter of is 25 cm, then the perimeter of is

42

In an isosceles triangle ABC if AC = BC and AB2 = 2AC2 , then =

43

is an isosceles triangle in which = 90° . If AC = 6 cm, then AB=

44

If in two triangles ABC and DEF, , , then which of the following not true?

45

In an isosceles triangle ABC, if AB = AC = 25 cm and BC = 14 cm, then the measure of altitude from A on BC is

46

In Fig. 4.242 the measures of and are respectively

47

In Fig. 4.243, the value of x for which is

48

In Fig. 4.244, if then CE =

49

In Fig. 4.245,. If CP = PD =11 cm and DR = RA = 3 cm. Then the values of x and y are respectively