4

Triangles

CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius

IMPORTANT POINTS 

If a line is drawn parallel to one side of a triangle (intersect the other two sides in two distinct point) divides other two sides in the same ratio.



If a line divides any two sides of a triangle in the same ratio, the line is parallel to third side.



If the corresponding sides of two triangles are proportional (i.e., in the same ratio) their corresponding angles are equal and hence the triangles are similar.



If one angle of a triangle is equal to one angle of the other and the sides including these angles are proportional, the triangles are similar.



If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.



In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two side.



In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.



If a line-segment drawn from the vertex of an angle of a triangle to it opposite side divides it in the ratio of the sides containing the angle, then the line segment bisects the angle.

Basic Proportionality theorem (Thales theorem) If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio. In  ABC A

if DE || BC then

AD AE = DB EC

D

E

B

C

Angle Bisector Theorem The internal bisector of an angle of a triangle divide opposite side internally in the ratio

of side containing the

angle. If Given then

1 = 2

BD AB = AC DC

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Exercise 4.1 1.

2.

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, equilateral)

(iv)

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

Give two different examples of pair of (i) similar figures. (ii) non-similar figures.

3.

State whether the following quadrilaterals are similar or not:

4.

If a line intersects sides AB and AC of a  ABC at D and E respectively and is parallel to BC, prove that AD AE .  AB AC

5.

ABCD is a trapezium with AB || DC. E and F are points on non-parallel sides AD and BC respectively such that EF is parallel to AB (see Fig.). Show that

AE BF .  ED FC

PS PT  and PST  PRQ . Prove that PQR is an isosceles triangle. SQ TR

6.

In fig,

7.

In Fig., (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).

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8.

E and F are points on the sides PQ and PR respectively of a  PQR. For each of the following cases, state whether EF || QR : (i)

PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

(ii)

PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

(iii)

PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm

AM AN  . AB AD

9.

In Fig, if LM || CB and LN || CD, prove that

10.

In Fig., DE || AC and DF || AE. Prove that

11.

In Fig., DE || OQ and DF || OR. Show that EF || QR.

12.

In Fig., A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.

13.

Using Theorem, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

14.

Using Theorem, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

15.

ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that

BF BE .  FE EC

AO CO  . BO DO

16.

The diagonals of a quadrilateral ABCD intersect each other at the point O such that

AO CO . Show  BO DO

that ABCD is a trapezium.

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Similar Triangles Two  s are said to be similar, if their corresponding angles are equal and their corresponding sides are proportional. (i)

The internal bisector of an angle divides the opposite side in the ratio of sides containing the angle.

(ii)

The line joining the mid points of any two sides of a triangle is parallel to the third side and equal to half of it.

(iii)

The diagonals of a trapezium divided each other proportionally.

(iv)

Ratio of areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides.

(v)

Ratio of area of two similar triangles is equal to the ratio of the squares of the corresponding altitudes.

(vi)

Ratio of areas of two similar triangles I equal to the ratio of the squares of the corresponding medians.

(vii) Ratio of areas of two similar triangles is equal to the ratio of the squares of the corresponding angle bisector segments? (viii) If the areas of two similar triangles are equal, then the triangles are congruent. (ix)

In two similar triangles, the ratio of two corresponding sides is equal to the ratio of corresponding medians.

(x)

In two similar triangles, the ratio of two corresponding sides is equal to the ratio of their corresponding heights.

Similarity of Triangle Two triangles are similar, if (i) Their corresponding angles are equal and (ii) Their corresponding sides are in the same ratio (or proportion). That is, in Δ ABC and Δ DEF, if (i)  A =  D,  B =  E,  C =  F (ii)

AB BC CA then the two triangles are similar.   DE EF FD

Similarity of triangle by (i) SAS AB BC  and  B =  E DE EF

(ii) By AAA (AA)  A =  D,  B = (iii) By SSS AB BC AC   DE EF DF

 E,  C=  F

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CLASS - X Mathematics

Triangles

Exercise 4.2 1.

In Fig., if PQ || RS, prove that  POQ ~  SOR.

2.

Observe Fig. and then find  P.

3.

In Fig., OA . OB = OC . OD. Show that  A =  C and  B =  D. C A

O D B

4.

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

5.

In Fig., CM and RN are respectively the medians of  ABC and  PQR. If  ABC ~  PQR, prove that: (i)  AMC ~  PNR (ii)

CM AB  RN PQ

(iii)  CMB ~  RNQ Q

N

R

A M

B

C

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P

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Triangles

6.

State which pairs of triangles in Fig. are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form :

7.

In Fig.,  ODC ~  OBA, ∠BOC = 125° and ∠CDO = 70°. Find  DOC,  DCO and  OAB.

8.

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

OA OB .  OC OD

QR QT  and  1 = ∠  2. Show that  PQS ~  TQR. QS PR

9.

In Fig.,

10.

S and T are points on sides PR and QR of  PQR such that ∠  P =  RTS. Show that  RPQ ~  RTS.

11.

In Fig., if  ABE   ACD, show that  ADE ~  ABC.

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12.

In Fig. altitudes AD and CE of  ABC intersect each other at the point P. Show that: (i)  AEP ~  CDP

(ii)  ABD ~  CBE

(iii)  AEP ~  ADB

(iv)  PDC ~  BEC

13.

E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that  ABE ~  CFB.

14.

In Fig., ABC and AMP are two right triangles, right angled at B and M respectively. Prove that: (i)  ABC ~  AMP

15.

(ii)

CA BC  PA MP

CD and GH are respectively the bisectors of ACB and EGF such that D and H lie on sides AB and FE of  ABC and  EFG respectively. If  ABC ~  FEG, show that: (i)

CD AC  GH FG

(ii)  DCB ~  HGE

(iii)  DCA ~  HGF

16.

In Fig., E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD  BC and EF  AC, prove that  ABD ~  ECF.

17.

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of  PQR (see Fig. 6.41). Show that  ABC ~  PQR.

18.

D is a point on the side BC of a triangle ABC such that  ADC =  BAC. Show that CA2 = CB.CD.

19.

Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that  ABC ~  PQR. 011-26925013/14 +91-9811134008 +91-9582231489

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20.

A vertical pole 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.

21.

If AD and PM are medians of triangles ABC and PQR, respectively where  ABC ~  PQR, prove that AB AD  . PQ PM

Area of similar Triangle (i) The ratio of area of two similar triangles is equal to ratio of square of their corresponding sides.

(ii)

AB AC BC   DE DF EF ar (ABC ) AB 2 BC 2 AC 2    then ar (DEF ) DE 2 EF 2 DF 2 The ratio of area of two similar triangle is equal to ratio of square of their corresponding median. if

 ABC ~  DEF ,

if

 ABC ~  DEF ,

AB AC BC   DE DF EF

ar (ABC ) AX 2  ar (DEF ) DY 2 The ratio of area of two similar triangles is equal to ratio of the square of their corresponding altitude. then

(iii)

AB AC BC   DE DF EF

if

 ABC ~  DEF ,

then

ar (ABC ) AL2  ar (DEF ) DM 2

Exercise 4.3 1.

In Fig. 6.43, the line segment XY is parallel to side AC of  ABC and it divides the triangle into two AX parts of equal areas. Find the ratio AB

2.

Let  ABC ~  DEF and their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC.

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3.

In the given figure, ABC and DEF are similar. The area of ABC is 9 sq. cm and the area of DEF is 16 sq. cm. If BC = 2.1 cm, find the length of EF. D

A

B 4.

C

F

E

In the given figure, ABC and DEF are similar, BC = 3 cm, EF = 4 cm and area of ABC = 54 cm2. D A Determine the area of DEF.

B

C

E

F

5.

Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2 CD, find the ratio of the areas of triangles AOB and COD.

6.

In Fig. 6.44, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that ar ( ABC) AO  ar ( DBC ) DO

7.

If the areas of two similar triangles are equal, prove that they are congruent.

8.

D, E and F are respectively the mid-points of sides AB, BC and CA of  ABC. Find the ratio of the areas of  DEF and  ABC.

9.

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.

10.

Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.

Tick the correct answer and justify :

11.

ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is (A) 2 : 1

12.

(B) 1 : 2

(C) 4 : 1

(D) 1 : 4

Sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio (A) 2 : 3

(B) 4 : 9

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(C) 81 : 16

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(D) 16 : 81

CLASS - X Mathematics

Triangles

Pythagoras Theorem In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. A right angled triangle ABC in which  B = 90o. Then AC2 = AB2 + BC2

Converse of Pythagoras Theorem In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the side is a right angle. A triangle ABC such that AC2 = AB2 + BC2 Then  B = 90o

Exercise 4.4 BC 2

BD . AD

1.

In Fig.,  ACB = 90° and CD  AB. Prove that

2.

A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches a window 6 m above the ground. Find the length of the ladder.

3.

In Fig., if AD  BC, prove that AB2 + CD2 = BD2 + AC2.

4.

BL and CM are medians of a triangle ABC right angled at A. Prove that 4 (BL 2 + CM2) = 5 BC2.

5.

O is any point inside a rectangle ABCD (see Fig.). Prove that OB2 + OD2 = OA2 + OC2.

6.

Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse. (i) 7 cm, 24 cm, 25 cm

AC 2

(ii) 3 cm, 8 cm, 6 cm



(iii) 50 cm, 80 cm, 100 cm

(iv) 13cm, 12cm, 5cm

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Triangles

7.

PQR is a triangle right angled at P and M is a point on QR such that PM  QR. Show that PM2 = QM . MR.

8.

In Fig., ABD is a triangle right angled at A and AC ⊥ BD. Show that

(i) AB2 = BC . BD

(ii) AC2 = BC . DC

(iii) AD2 = BD . CD

9.

ABC is an isosceles triangle right angled at C. Prove that AB 2 = 2AC2.

10.

ABC is an isosceles triangle with AC = BC. If AB2 = 2 AC2, prove that ABC is a right triangle.

11.

ABC is an equilateral triangle of side 2a. Find each of its altitudes.

12.

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

13.

In Fig., O is a point in the interior of a triangle ABC, OD  BC, OE  AC and OF  AB. Show that

(i) OA2 + OB2 + OC2 – OD2– OE2 – OF2 = AF2 + BD2 + CE2, BF2.

(ii) AF2 + BD2 + CE2 = AE2 + CD2+

14.

A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.

15.

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?

16.

An aeroplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, 1 another aeroplane leaves the same airport and flies due west at a speed of 1200 km per 1 hour. How far 2 apart will be the two planes after hours?

17.

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

18.

D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that AE2 + BD2 = AB2 + DE2.

19.

The perpendicular from A on side BC of a  ABC intersects BC at D such that DB = 3 CD (see Fig.). Prove that 2 AB2 = 2 AC2 + BC2.

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1 BC. Prove that 9 AD2 = 7 AB2. 3

20.

In an equilateral triangle ABC, D is a point on side BC such that BD =

21.

In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

22.

Tick the correct answer and justify : In  B is: (A) 120°

3 cm, AC = 12 cm and BC = 6 cm. The angle

(B) 60°

(C) 90°

(D) 45°

Exercise 4.5 QS PQ .  SR PR

1.

In Fig., PS is the bisector of  QPR of  PQR. Prove that

2.

In Fig., D is a point on hypotenuse AC of  ABC, DM  BC and DN  AB. Prove that : (i) DM2 = DN . MC

(ii) DN2 = DM . AN

3.

In Fig., ABC is a triangle in which  ABC > 90° and AD  CB produced. Prove that AC2 = AB2 + BC2 + 2 BC . BD.

4.

In Fig., ABC is a triangle in which  ABC < 90° and AD  BC. Prove that AC2 = AB2 + BC2 –2 BC . BD.

5.

In Fig., AD is a median of a triangle ABC and AM  BC. Prove that :

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 BC  (i) AC = AD + BC . DM +    2  2

2

2

(iii) AC2 + AB2 = 2 AD2 +

 BC  (ii) AB = AD – BC . DM +    2  2

2

2

1 BC2 2

6.

Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.

7.

In Fig., two chords AB and CD intersect each other at the point P. Prove that:

(i)  APC ~  DPB

8.

(ii) AP . PB = CP . DP

In Fig., two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i)  PAC ~  PDB

9.

(ii) PA . PB = PC . PD

In Fig., D is a point on side BC of  ABC such that

 BAC.

10.

BD AB . Prove that AD is the bisector of  CD AC

Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, how much string does she have out (see Fig. 6.64)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?

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ANSWER Exercise 4.1 1. (i) similar

(ii) Similar

(iii) Equivalent

2. (i) All circle and all square are similar rectangle are not similar. 8. (i) No

(ii) Yes

(iv) (a) equal

(b) Proportional

(ii) a circle and a square are not similar and Rhombus and a

(iii) Yes

Exercise 4.2 (2) P = 4

(4) l = 1.6metre

6.(i) Yes, AAA, ABC ~ PQR

(ii) Yes, SSS, ABC ~ QRP

(iv) Yes, SAS, MNL ~ QPR

(v) No

(iii) No

(vi) Yes, AA, DEF ~ PQR

7. 55, 55, 55 20. 42 metre Exercise 4.3 (2) 11.2cm

(3) 2.8cm

(11) c

(12) d

(4) 96cm2

(5) 4 : 1

8. 1 : 4

Exercise 4.4 (2) l = 6.5m (6) (i) Yes H = 25cm

(ii) No

(iii) No

(iv) Yes, 13cm

(11) a 3

(14) 6m

(15) 6 7m

(16) 300 61km

(17) 13m

(22) c Exercise 4.5 (10) 3m, 2.79m

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