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[49] 011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
CHAPTER
4We are Starting from a Point but want to Make it a Circle of Infinite Radius
Triangles
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
if DE || BC
then DB
AD =
EC
AE
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 1 = 2
then AC
AB =
DC
BD
A
D
B C
E
[50] 011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
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, 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)
2. 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
AC
AE
AB
AD .
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 FC
BF
ED
AE .
6. In fig, TR
PT
SQ
PS and PRQPST . Prove that PQR is an isosceles triangle.
7. In Fig., (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).
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NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
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
9. In Fig, if LM || CB and LN || CD, prove that AD
AN
AB
AM .
10. In Fig., DE || AC and DF || AE. Prove that EC
BE
FE
BF .
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
DO
CO
BO
AO .
16. The diagonals of a quadrilateral ABCD intersect each other at the point O such that DO
CO
BO
AO . Show
that ABCD is a trapezium.
[52] 011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
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) FD
CA
EF
BC
DE
AB then the two triangles are similar.
Similarity of triangle by
(i) SAS
AB BC
DE EF and B = E
(ii) By AAA (AA)
A =D, B = E, C= F
(iii) By SSS
AB BC AC
DE EF DF
[53] 011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
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.
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) PQ
AB
RN
CM
(iii) CMB ~ RNQ
A
B
C
D O
A
B C
M
Q R N
P
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NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
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 OD
OB
OC
OA .
9. In Fig., PR
QT
QS
QR and 1 = ∠ 2. Show that PQS ~ TQR.
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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NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
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 (ii) MP
BC
PA
CA
15. 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) FG
AC
GH
CD
(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.
[56] 011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
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
PM
AD
PQ
AB .
Area of similar Triangle
(i) The ratio of area of two similar triangles is equal to ratio of square of their corresponding sides.
if ABC ~ DEF , EF
BC
DF
AC
DE
AB
then 2
2
2
2
2
2
)(
)(
DF
AC
EF
BC
DE
AB
DEFar
ABCar
(ii) The ratio of area of two similar triangle is equal to ratio of square of their corresponding median.
if ABC ~ DEF , EF
BC
DF
AC
DE
AB
then 2
2
)(
)(
DY
AX
DEFar
ABCar
(iii) The ratio of area of two similar triangles is equal to ratio of the square of their corresponding
altitude.
if ABC ~ DEF , EF
BC
DF
AC
DE
AB
then 2
2
)(
)(
DM
AL
DEFar
ABCar
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
parts of equal areas. Find the ratio AB
AX
2. Let ABC ~ DEF and their areas be, respectively, 64 cm2 and 121 cm
2. If EF = 15.4 cm, find BC.
[57] 011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
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.
4. In the given figure, ABC and DEF are similar, BC = 3 cm, EF = 4 cm and area of ABC = 54 cm2.
Determine the area of DEF.
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
DO
AO
DBCar
ABCar
)(
)(
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 (B) 1 : 2 (C) 4 : 1 (D) 1 : 4
12. 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 (C) 81 : 16 (D) 16 : 81
A
D
B
C
E
F
D
A
B
C
E
F
[58] 011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
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 = AB
2 + BC
2
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 = AB
2 + BC
2
Then B = 90o
Exercise 4.4
1. In Fig., ACB = 90° and CD AB. Prove that AD
BD
AC
BC
2
2
.
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 + CD
2 = BD
2 + AC
2.
4. BL and CM are medians of a triangle ABC right angled at A. Prove that 4 (BL2 + CM
2) = 5 BC
2.
5. O is any point inside a rectangle ABCD (see Fig.). Prove that OB2 + OD
2 = OA
2 + OC
2.
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 (ii) 3 cm, 8 cm, 6 cm (iii) 50 cm, 80 cm, 100 cm
(iv) 13cm, 12cm, 5cm
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NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
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) AC
2 = BC . DC (iii) AD
2 = BD . CD
9. ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC
2.
10. ABC is an isosceles triangle with AC = BC. If AB2 = 2 AC
2, 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 + OB
2 + OC
2 – OD
2– OE
2 – OF
2 = AF
2 + BD
2 + CE
2, (ii) AF
2 + BD
2 + CE
2 = AE
2 + CD
2+
BF2.
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,
another aeroplane leaves the same airport and flies due west at a speed of 1200 km per 12
1hour. How far
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 + BD
2 = AB
2 + DE
2.
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 AC
2 + BC
2.
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NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
20. In an equilateral triangle ABC, D is a point on side BC such that BD = 3
1BC. Prove that 9 AD
2 = 7 AB
2.
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 3 cm, AC = 12 cm and BC = 6 cm. The angle
B is:
(A) 120° (B) 60° (C) 90° (D) 45°
Exercise 4.5
1. In Fig., PS is the bisector of QPR of PQR. Prove that PR
PQ
SR
QS .
2. In Fig., D is a point on hypotenuse AC of ABC, DM BC and DN AB. Prove that :
(i) DM2 = DN . MC (ii) DN
2 = DM . AN
3. In Fig., ABC is a triangle in which ABC > 90° and AD CB produced. Prove that AC2 = AB
2 + BC
2
+ 2 BC . BD.
4. In Fig., ABC is a triangle in which ABC < 90° and AD BC. Prove that AC2 = AB
2 + BC
2 –2 BC .
BD.
5. In Fig., AD is a median of a triangle ABC and AM BC. Prove that :
[61] 011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
(i) AC2 = AD
2 + BC . DM +
2
2
BC
(ii) AB2 = AD
2 – BC . DM +
2
2
BC
(iii) AC2 + AB
2 = 2 AD
2 +
2
1BC
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 (ii) AP . PB = CP . DP
8. 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 (ii) PA . PB = PC . PD
9. In Fig., D is a point on side BC of ABC such that AC
AB
CD
BD . Prove that AD is the bisector of
BAC.
10. 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?
[62] 011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
TrianglesCLASS - XMathematics
ANSWER
Exercise 4.1
1. (i) similar (ii) Similar (iii) Equivalent (iv) (a) equal (b) Proportional
2. (i) All circle and all square are similar (ii) a circle and a square are not similar and Rhombus and a
rectangle are not similar.
8. (i) No (ii) Yes (iii) Yes
Exercise 4.2
(2) P = 4 (4) l = 1.6metre
6.(i) Yes, AAA, ABC ~ PQR (ii) Yes, SSS, ABC ~ QRP (iii) No
(iv) Yes, SAS, MNL ~ QPR (v) No (vi) Yes, AA, DEF ~ PQR
7. 55, 55, 55
20. 42 metre
Exercise 4.3
(2) 11.2cm (3) 2.8cm (4) 96cm2 (5) 4 : 1 8. 1 : 4
(11) c (12) d
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