Assignments For Class 10 Mathematics Triangles

Assignments Assignments for Class 10

Assignments for Class 10 Mathematics Triangles have been developed for Standard 10 students based on the latest syllabus and textbooks applicable in CBSE, NCERT and KVS schools. Parents and students can download the full collection of class assignments for class 10 Mathematics Triangles from our website as we have provided all topic wise assignments free in PDF format which can be downloaded easily. Students are recommended to do these assignments daily by taking printouts and going through the questions and answers for Grade 10 Mathematics Triangles. You should try to do these test assignments on a daily basis so that you are able to understand the concepts and details of each chapter in your Mathematics Triangles book and get good marks in class 10 exams.

Question. In the given figure, AB || CD then the value of ‘x’ is equal to   

Assignments For Class 10 Mathematics Triangles

(A) 440
(B) 880
(C) 800
(D)1000

Answer

B

In the adjoining figure ABC is a triangle; P is an interior point in it. Three lines are drawn through the point P, parallel to three sides as shown in figure. The triangle is divided into six parts. The areas as 3 smaller triangles are 4, 9 and 16 units, then the area of triangle ABC is…….     

(A) 64
(B) 81
(C) 42
(D) 65

Answer

B

Question. If ∠BAE = ∠ECD then ΔABD and ΔCDEwill be

Assignments For Class 10 Mathematics Triangles

(A) congruent
(B) similar
(C) right angle triangle
(D) cannot say

Answer

B

Question. If ΔAEB and ΔDCAboth are right angled triangle then which of the following is correct?   

Assignments For Class 10 Mathematics Triangles

(A) CD/EB DA/AC
(B) AB/AC = AD/DC
(C) EA/AC = EB/CD
(D) none of these

Answer

C

ΔABC is an equilateral triangle, we have BD = EG = DF = DE = EC, then the ratio of the area of the portion to area of ΔABC is:   

Assignments For Class 10 Mathematics Triangles

(A) 4/11
(B) 7/9
(C) 5/12
(D) 6/7

Answer

B

Question. In the given figure, AB divides ∠DACin the ratio 1 : 3 and AB = DB. Find the value of x.   

Assignments For Class 10 Mathematics Triangles

(A) 900
(B) 800
(C) 700
(D) 850

Answer

A

Question. In the quadrilateral ABCD 90 , ∠A = ∠C= 900 ,AE = 5cmand BE = 12cm and AC = 21 cm. If DF = x, then the value of x is: 

Assignments For Class 10 Mathematics Triangles

(A) 6 (2/3) cm
(B) 10/7 cm
(C) 11 cm
(D) 13 cm

Answer

A

Question. In the given figure, o ∠A =1000 and AB = AC, find ∠B and ∠C.   

Assignments For Class 10 Mathematics Triangles

(A) 400 ,400
(B) 600 ,200
(C) 450 ,350
(D) 250 ,550

Answer

A

Question. In the diagram ABCD is a rectangle with AE = EF =FB, the ratio of the areas of triangle CEF and that of rectangle ABCD is 

Assignments For Class 10 Mathematics Triangles

(A) 1 : 6
(B) 1 : 8
(C) 1 : 9
(D) 1 : 10

Answer

A

Question. Which of the following is not a criterion the congruence of triangles?   
(A) SAS
(B) SSA
(C) ASA
(D) SSS

Answer

B

Question. In the given figure, The measure of ∠B’A’C’ is   

Assignments For Class 10 Mathematics Triangles

(A) 500
(B) 600
(C) 700
(D) 800

Answer

B

Question. In the figure, the area of square ABCD is 4 cm2 and E is midpoint of AB; F, G, H and K are the mid points of DE, CF, DG and CH respectively. The area of triangle KDC is   

Assignments For Class 10 Mathematics Triangles

(A) 1/4 cm2
(B) 1/8 cm2
(C) 1/16 cm2
(D) 1/32 cm2

Answer

B

Question. In the given figure ABCD is a rectangle and all measurement is in centimeters. Find the area of the shaded region   

Assignments For Class 10 Mathematics Triangles

(A) 240 cm2
(B) 205 cm2
(C) 105 cm2
(D) 95 cm2

Answer

C

Question. If A, B and C are mid-points of DE, EF and ED of ΔDEF then find the ratio of area of ΔABC and ΔDEF .   
(A) 1 : 4
(B) 4 : 1
(C) 3 : 2
(D) 2 : 3

Answer

B

Question. Which of the following is correct?     

Assignments For Class 10 Mathematics Triangles

(A) AC = AB
(B) AC = BC
(C) DC = BD
(D) BC = AC

Answer

C

Question. In the given figure, X is point in the interior of square ABCD.   
AXYZ is also a square. If DY = 3cm, AZ = 2cm then length of BY is:
(A) 5 cm
(B) 6 cm
(C) 7 cm
(D) 8 cm

Answer

C

Question. In a square PQRS, A and B are two points on Ps and sR such that PA = 2As and RB = 2Bs. If PQ = 6, the area of the triangle ABQ is.   
(A) 5sq. units
(B) 10sq. units
(C) 15sq. units
(D) None of these

Answer

B

Question. If ΔPQR ≅ ΔEFD, then ED = ?   
(A) PQ
(B) QR
(C) PR
(D) none of these

Answer

C

Question. In the given figure, AB = AC, AD is the median to base BC. Then, ∠BAD= ?   

Assignments For Class 10 Mathematics Triangles

(A) 550
(B) 700
(C) 350
(D) 1100

Answer

A

The ratio of the area of A to that of C is 16 : 27 and the ratio of B to the area of C is 1 : 3. Find the ratio of the area of A to that of 

Assignments For Class 10 Mathematics Triangles

(A) 4 : 5
(B) 5 : 4
(C) 1 : 1
(D) None of these

Answer

A

Question. In ΔABC, o ∠B=∠C = 45 . Which is the longest side?   
(A) AC
(B) AB
(C) BC
(D) none of these

Answer

C

Question. In ΔABC, if o ∠A = 500  and  ∠B = 600 , determine the shortest and largest sides of the triangle.   
(A) BC, AB
(B) AB, BC
(C) AC, BC
(D) none of these

Answer

A

Question. The fig. below has been obtained by folding a rectangle. The total area of the figure is 144 cm2. Had the rectangle not been folded, the current overlapping part would have been a square. What would have been the total area of the original unfolded rectangle?     

Assignments For Class 10 Mathematics Triangles

(A) 162m2
(B) 140cm2
(C) 142mm2
(D) 162cm2

Answer

D

Question. ABCD is a parallelogram, if the two diagonals are equal, find the measure of ΔABC.   
(A) 50
(B) 60
(C) 90
(D) 100

Answer

C

Question. In the given figure, AC is the bisector of ∠BAD. Then CD = ?   

Assignments For Class 10 Mathematics Triangles

(A) 2cm
(B) 3 cm
(C) 4 cm
(D) 5cm

Answer

C

Question. In ΔABCand ΔDEF such that ΔABC ≅ ΔFDE, and AB = 5 cm, ∠B = 400 , ∠A = 800 . Which of the following is true? 
(A) DF = 5 cm, ∠F = 600
(B) DE = 5 cm, ∠E = 600
(C) DF = 5 cm, ∠E = 600
(D) DE = 5 cm,∠D = 400

Answer

C

Question. If XYZ is a triangle where ∠Z = 900 . If L is the mid-point of YZ then 
(A) XY2 + 4XL2 = 3XZ2
(B) XY2 + 3XZ2 = 4XL2
(C) XY2 + XZ2 = XL2
(D) none of these

Answer

B

Question. If hypotenuse LM is common for both the triangles i.e., ΔKLMand ΔLMN then   

Assignments For Class 10 Mathematics Triangles

(A) KX x XM = LX x LM
(B) KX x KL  = LM x MX
(C) KX x XM = LX x XN
(D) none of these 

Answer

C

Question. In the triangles ABC and PQR, three equality relations between some parts are as follows: 
AB = PQ, ∠B= ∠P , BC = PR Congruence conditions apply:
(A) SAS
(B) ASA
(C) SSS
(D)RHS

Answer

A

Question. In an isosceles triangle, if the vertex angle is twice the sum of the base angles, then the measure of the vertex angle of the triangle is   
(A) 1000
(B) 1200
(C) 1100
(D) 1300

Answer

A

VERY SHORT ANSWER TYPE QUESTIONS

Question. If the ratio of the corresponding medians of two similar triangles is 7 : 5, then what is the ratio of their corresponding sides?

Answer

7 : 5

Question. The lengths of sides of a triangle are 12 cm, 16 cm and 21 cm. The bisector of the greatest angle divides the opposite side into two parts. Find the length of these two parts.

Answer

9 cm, 12 cm

Question. In figure, find AD.   

Assignments For Class 10 Mathematics Triangles
Answer

1.5 cm

Question. Is the triangle with sides 9 cm, 41 cm, 40 cm a right angled triangle?

Answer

yes

Question. If ΔABC ~ ΔPQR such that ar (ΔABC) = 81 cm2 and ar (ΔPQR) = 121 cm2. If QR = 22 cm, find BC.

Answer

18 cm

Question. In figure, ΔXMN ~ ΔXYZ, what is the measure of ΔXZY?   

Assignments For Class 10 Mathematics Triangles
Answer

40°

Question. In figure, ∠ADB = ∠BAC = 90°. What is the value of x?   

Assignments For Class 10 Mathematics Triangles
Answer

24 cm

Question. The perimeters of two similar triangles ABC and PQR are respectively 30 cm and 24 cm. If PQ = 8 cm, find AB.

Answer

10 cm

Question. In figure, express x in the terms of a, b and c.   

Assignments For Class 10 Mathematics Triangles
Answer

ac/b + c

Question. If ΔABC ~ ΔPQR and their corresponding altitudes are in the ratio 7 : 8, then what is the ratio of the area of ΔPQR to that of area of ΔABC?

Answer

64 : 49

Question. In figure, find x.   

Assignments For Class 10 Mathematics Triangles
Answer

8 cm

Question. In figure, DE || BC. If AD/DB = 4/3 , then what is the value of AE/AC ?   

Assignments For Class 10 Mathematics Triangles
Answer

4/7

Question. Two poles of height 6 m and 11 m are standing 12 m apart. What is the distance between their tops?

Answer

13 m

Question. If ΔABC ~ ΔXYZ and the ratio of area of ΔABC to that of area of ΔXYZ is 25 : 49, then what is the ratio of their corresponding medians?

Answer

5 : 7

Question. In figure, ΔABC and ΔDEC are the right triangles with ∠B = ∠E = 90°, find BE.   

Assignments For Class 10 Mathematics Triangles
Answer

23 cm

Question. If the ratio of the corresponding sides of two similar triangles is 3 : 5, then what is the ratio of their corresponding heights?

Answer

3 : 5

Question. The area of the two similar triangles are in the ratio 81 : 121. What is the ratio of their corresponding sides?

Answer

9 : 11

Question. In figure, DE || BC, what is the value of x? 

Assignments For Class 10 Mathematics Triangles
Answer

25/3 cm

Question. If ΔABC ~ ΔDEF such that BC = 5 cm, EF = 3 cm and ar (ΔDEF) = 36 cm2, find the area of ΔABC.

Answer

100 cm2

Assignment for Class 10 Mathematics Triangles Set A
Assignment for Class 10 Mathematics Triangles Set B
Assignment for Class 10 Mathematics Triangles Set C
Assignment for Class 10 Mathematics Triangles Set D
Assignment for Class 10 Mathematics Triangles Set E
Assignment for Class 10 Mathematics Triangles Set F
Assignment for Class 10 Mathematics Triangles Set G
Assignment for Class 10 Mathematics Triangles Set H
Assignment for Class 10 Mathematics Triangles Set I
Assignment for Class 10 Mathematics Triangles Set J
Assignment for Class 10 Mathematics Triangles Set K
Assignment for Class 10 Mathematics Triangles Set L
Assignment for Class 10 Mathematics Triangles Set M

Triangles

Similar figures : Two polygons of the same number of sides are similar, if :
i) Their corresponding angles are equal.
ii) Their corresponding sides are in the same ratio (or proportion)
Properties for similar triangles
Area of similar triangles
A number of techniques can be applies in accordance with the similarity theorems to easily compute areas of similar triangles.
“The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides”

Assignments For Class 10 Mathematics Triangles

Properties of right angle triangle F

Assignments For Class 10 Mathematics Triangles

i) Pythagoras theorem AC2=AB2+BC2
ii) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other.
Two triangles are similar, if :
i) Their corresponding angle are equal ( AAA-criteria).
ii) Their corresponding sides are in the same ratio (SSS-criteri ).
iii) If one angle of a triangle is equal to the one angle of other triangle and sides including these angles angle are in proportion (SAS-criteri ).
Theorems used to prove similarity of triangles are :

I. Basic Proportionality Theorem (B.P.T.) (Theorem)
In a triangle, a line drawn parallel to one side, to intersect the other sides in distinct points, divides the sides in Δ the same ratio. 
In ABC,if DE ∥ BC then F

Assignments For Class 10 Mathematics Triangles
Assignments For Class 10 Mathematics Triangles

II. Converse of Basic proportionality Theorem
If a line divides any two sides of a triangle in the same ratio, the line is parallel to the third side. In Δ PA/SA =PT/TR, then AT ∥ QR.F

Assignments For Class 10 Mathematics Triangles

Solved Problems on Class 10 Triangles

Question. In Fig.4.18, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ^ BC and EF ^ AC, prove that ΔABD ~ΔECF. 

Assignments For Class 10 Mathematics Triangles

Sol. We have, ∠B = ∠C [ ABC is an isosceles triangle with AB = AC]
Now, in ΔABD and ΔECF
  ∠ABD = ∠ECF           [ ∠B = ∠C]
  ∠ADB = ∠EFC = 90° [ AD ^ BC and EF ^ AC]
 ΔABD ~ ΔECF         (By AA criterion of similarity)

Question. In Fig. 4.12, AO/OC=BO/OD = 1/2 and AB = 5 cm. Find the value of DC.

Assignments For Class 10 Mathematics Triangles

Sol. In ΔAOB and ΔCOD, we have
∠AOB = ∠COD            [Vertically opposite angles]
AO/OC=BO/OD           [Given]
So, by SAS criterion of similarity, we have ΔAOB ~ΔCOD
AO/OC=BO/OD=AB/DC
     1/2= 5/DC         [Q AB = 5 cm]
    DC =10 cm

Question. 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/OC=OB/OD.
Sol. Given: ABCD is a trapezium in which AB||DC.
To prove: OA/OC=OB/OD
Proof: In DOAB and DODC, we have
∠OAB = ∠OCD (Alternate angles)
∠AOB = ∠DOC (Vertically opposite angles)
∠ABO = ∠ODC (Alternate angles)
∴ ΔOAB ~ΔOCD (By AA criterion of similarity)
Hence, OA/OC=OB/OD

Question. ABCD is a trapezium in which AB||DC and its diagonals intersect each other at the point O. Show that
AO/BO=CO/DO.

Assignments For Class 10 Mathematics Triangles

Sol. Given: ABCD is a trapezium, in which AB||DC and its diagonals intersect each other at the point O.
To prove: AO/BO=CO/DO
Construction: Through O, draw OE|| AB i.e., OE||DC.
Proof: In ΔADC, we have OE||DC (Construction)
 By Basic Proportionality Theorem, we have
AE/ED=AO/CO                                         …(i)
Now, in DABD, we have OE|| AB (Construction)
 By Basic Proportionality Theorem, we have
ED/AE=DO/BO  AE/ED=BO/DO               …(ii)
From (i) and (ii), we have
AO/CO=BO/DO  AO/BO=CO/DO

Question. S and T are points on sides PR and QR of DPQR such that ∠P = ∠RTS. Show that ∠RPQ ~DRTS. 

Assignments For Class 10 Mathematics Triangles

Sol. In DRPQ and DRTS, we have
∠RPQ = ∠RTS            (Given)
∠PRQ = ∠TRS = ∠R   (Common)
 ΔRPQ ~ΔRTS          (By AA criterion of similarity.)

Question. State which pairs of triangles in the following figures 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.

Assignments For Class 10 Mathematics Triangles

Sol. (i) In ΔABC and ΔPQR, we have
AB/QR= 2/4=1/2
AC/PQ= 3/6=1/2
BC/PR= 2.5/5=25/50=1/2
Hence, AB/QR=AC/PQ=BC/PR
ΔABC ~ΔQRP by SSS criterion of similarity.
(ii) In ΔLMP and ΔDEF, we have
LP/DF=3/6=1/2, MP/DE=2/4=1/2,  LM/EF=2.7/5.
Hence, LP/DF=MP/DE≠LM/EF
∴ ΔLMP is not similar to ΔDEF.
(iii) In ΔNML and ΔPQR, we have
∠M = ∠Q = 70°
Now, MN/PQ= 2.5/6=5/12.
And ML/QR= 5/10=1/2
Hence MN/PQ≠ML/QR
∴ ΔNML is not similar to ΔPQR because they do not satisfy SAS criterion of similarity.

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

Assignments For Class 10 Mathematics Triangles

Sol. In ΔABC and ΔDAC, we have
ÐBAC = ∠ADC (Given)
and ∠C = ∠C (Common)
 ΔABC ~ ΔDAC (By AA criterion of similarity)
⇒ AB/DA=BC/AC=AC/DC
⇒ CB/CA=CA/CD
⇒ CA2 = CB ≠ CD

Question. E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ΔABE ~ΔCFB.

Assignments For Class 10 Mathematics Triangles

Sol. In ΔABE and ΔCFB, we have
  ∠AEB = ∠CBF        (Alternate angles)
  ∠A = ∠C               (Opposite angles of a parallelogram)
∴ ΔABE ~ΔCFB        (By AA criterion of similarity)

Question. 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.

Assignments For Class 10 Mathematics Triangles

Sol. Let AB be a vertical pole of length 6m and BC be its shadow and DE be tower and EF be its shadow. Join AC and DF.
Now, in ΔABC and ΔDEF, we have
∠B = ∠E = 90°
∠C = ∠F (Angle of elevation of the Sun)
 ΔABC ~ΔDEF (By AA criterion of similarity)
Thus, AB/DE=BC/EF
6/h= 4/28             (Let DE = h)
6/h=1/7
h = 42
Hence, height of tower, DE = 42 m

Question. Is the triangle with sides 12 cm, 16 cm and 18 cm a right triangle? Give reason.
Sol. Here, 122 + 162 = 144 + 256 = 400 ¹ 182
∴ The given triangle is not a right triangle.

3. Prove that the area of an equilateral triangle described on a side of a right-angled isosceles triangle is half the area of the equilateral triangle described on its hypotenuse.

Assignments For Class 10 Mathematics Triangles

Sol.Given: A ΔABC in which ÐABC = 90° and AB = BC. ΔABD and ΔACE are equilateral triangles.
To Prove: ar(ΔABD) = 1/2 × ar(ΔCAE)
Proof: Let AB = BC = x units.
 hyp. CA = x2+x2 = x2 units.
Each of the ΔABD and ΔCAE being equilateral, each angle of each one of them is 60°.
 ΔABD ~ ΔCAE
But, the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
 ar( ABD)/ar (CAE)=AB2/CA2 = x2/(x2)2 = x2/2x2=1/2
Hence, ar (ΔABD) = 1/2 × ar (ΔCAE)

Question. Two sides and the perimeter of one triangle are respectively three times the corresponding sides and the perimeter of the other triangle. Are the two triangles similar? Why?
Sol. Since the perimeters and two sides are proportional
 the third side is proportional to the third side.
i.e., the two triangles will be similar by SSS criterion.

Question. In Fig. 4.17, ABC and AMP are two right triangles right-angled at B and M respectively. Prove that:
(i) ΔABC~ΔAMP (ii) CA/PA=BC/MP

Assignments For Class 10 Mathematics Triangles

Sol. (i) In DABC and DAMP, we have
∠ABC =∠AMP = 90°            (Given)
And, ∠BAC = ∠MAP             (Common angle)
 ΔABC ~ΔAMP                   (By AA criterion of similarity)
(ii) As DABC ~DAMP            (Proved above)
 CA/PA=BC/MP               (Sides of similar triangles are proportional)

Question. 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. 

Assignments For Class 10 Mathematics Triangles

Sol. Let ABCD be a square and ΔBCE and ΔACF have been drawn on side BC and the diagonal AC respectively.
To prove: area (ΔBCE) = 1/2 area (ΔACF)
Proof: Since DBCE and DACF are equilateral triangles
ΔBCE ~ΔACF                                  (by AAA criterion of similarity)
 area(ΔBCE)/area(ΔACF)=BC2/AC2
 area(ΔBCE)/area(ΔACF) =BC2/BC2   [Q Diagonal = 2 side, AC = 2BC]
 area(ΔBCE)/area(ΔACF) = 1/2
 area (ΔBCE) = 1 area (ΔACF)

Question. In Fig. 4.21, ABCD is a trapezium with AB||DC. If ΔAED is similar to DBEC, prove that AD = BC.

Assignments For Class 10 Mathematics Triangles

Sol. In ΔEDC and ΔEBA, we have
∠1 = ∠2 [Alternate angles]
∠3 = ∠4 [Alternate angles]
and ∠CED = ∠AEB [Vertically opposite angles]
ΔEDC ~ΔEBA [By AA criterion of similarity]
ED/EB=EC/EA
ED/EC=EB/EA …(i)
It is given that ΔAED ~ΔBEC
ED/EC=EA/EB=AD/BC …(ii)
From (i) and (ii), we get
EB/EA=EA/EB (EB)2 = (EA)2 EB = EA
Substituting EB = EA in (ii), we get
EA/EA=AD/BC
AD/BC AD = BC

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

Assignments For Class 10 Mathematics Triangles

Sol. Given: Two triangles ABC and DEF, such that
DABC ~ΔDEF and area (ΔABC) = area (ΔDEF)
To prove: ΔABC ∼ ΔDEF
Proof: ΔABC ~ΔDEF
 ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
and AB/DE=BC/EF=AC/DF
Now, ar (ΔABC) = ar (ΔDEF) (Given)
 ar(ABC )/ar(DEF )=1 …(i)
and AB/DE=BC/EF=AC/DF ar(ABC )/ar(DEF ) ( ΔABC ~ΔDEF) …(ii)
From (i) and (ii), we have
AB/DE=BC/EF=AC/DF
 AB/DE=BC/EF=AC/DF
 AB = DE,       BC = EF,       AC = DF
Hence, ΔABC ∼ ΔDEF (By SSS criterion of congruency)

Question. Let ΔABC~ΔDEF and their areas be respectively 64 cm2 and 121 cm2. If EF =15× 4 cm, find BC.

Assignments For Class 10 Mathematics Triangles
Assignments For Class 10 Mathematics Triangles

Sol. We have,
area of ABC/ area of DEF =BC2/EF2 (as ΔABC ~ΔDEF)
 64/=121 = BC2/EF2
 64/121=BC2/(15-4)
 BC/15.4=8/11  BC = 8/11×15.4 = 11.2

Question. In Fig. 4.23, ABD is a triangle right-angled at A and AC ^ BD. Show that
(i) AB2 = BC . BD (ii) AD2 = BD . CD (iii) AC2 = BC . DC 

Assignments For Class 10 Mathematics Triangles

Sol. Given: ABD is a triangle right-angled at A and AC ^ BD.
To prove: (i) AB2 = BC . BD
(ii) AD2 = BD . CD
(iii) AC2 = BC . DC
Proof: (i) In ΔACB and ΔDAB, we have
∠ACB = ∠DAB = 90°
∠ABC = ∠DBA = ∠B (Common)
 ΔACB ~ ΔDAB (By AA criterion of similarity)
 BC/AB=AB/DB
AB2 = BC x BD
(ii) In ΔACD and ΔBAD, we have
∠ACD = ∠BAD = 90°
∠CDA = ∠BDA = ∠D (Common)
 ΔACD ~ΔBAD (By AA criterion of similarity)
 AD/BD=CD/AD
AD2 = BD x CD
(iii) We have DACB ~DDAB
 ΔBCA ~ΔBAD …(i)
and ΔACD ~ΔBAD …(ii)
From (i) and (ii), we have
ΔBCA ~ΔACD
 BC/AC=AC/DC
 AC2 = BC. DC

Question. If AD and PM are medians of triangles ABC and PQR respectively, where ΔABC ~ΔPQR, prove that
AB/PQ=AD/PM

Assignments For Class 10 Mathematics Triangles

Sol. In ΔABD and ΔPQM, we have∠B = ∠Q (Q ΔABC ~ΔPQR) …(i)
AB/PQ=BC/QR (Q ΔABC ~ΔPQR)
 AB/PQ=(1/2)BC/(1/2)QR
 AB/PQ=BD/QM = [Since AD and PM are the medians of DABC and DPQR respectively] …(ii)
From (i) and (ii), it is proved that
ΔABD ~ΔPQM (By SAS criterion of similarity)
AB/PQ=BD/QM=AD/PM  AB/PQ=AD/PM

Question. ABC is a triangle in which AB = AC and D is a point on AC such that BC2 = AC x CD. Prove that BD = BC.

Assignments For Class 10 Mathematics Triangles

Sol. Given: DABC in which AB = AC and D is a point on the side AC such that BC2 = AC x CD
To prove: BD = BC
Construction: Join BD
Proof: We have, BC2 = AC x CD
Þ BC/CD=AC/BC …(i)
Thus, in DABC and DBDC, we have
AC/BC =BC/CD [From (i)]
and ∠C = ∠C [Common]
 ΔABC ~ΔBDC
Þ AB/BD=BC/CD …(ii)
From (i) and (ii), we get
AC/BC=AB/BD
 BD = BC (Q AB = AC)

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

Assignments For Class 10 Mathematics Triangles

Sol. Let ΔABC and ΔPQR be two similar triangles. AD and PM are the medians of DABC and DPQR respectively.
To prove:
ar(ΔABC )/ar(ΔPQR)=AD2/PM2
Proof: Since DABC ~DPQR
∴ ar(ΔABC )/ar(ΔPQR)=AB2/PQ2                                                      …(i)
In ΔABD and ΔPQM
AB/PQ=BD/QM                                          (Q AB/PQ=BC/QR=BC/QR)
and ∠B = ∠Q                                           (Q ΔABC ~ΔPQR)
Hence, ΔABD ~ ΔPQM (By SAS Similarity criterion)
 AB/PQ=AD/PM                                                                         …(ii)
From (i) and (ii), we have
ar(ΔABC )/ar(ΔPQR)=AD2/PM2

2. State whether the following statements are true or false. Justify your answer.
(i) If DE/PQ=EF/PR and ÐD= ÐQ, then DDEF ~DPQR.
(ii) P and Q are the points on the sides DE and DF of a triangle DEF such that DP = 4 cm, PE= 14 cm, DQ = 6 cm and DF= 21 cm. Then PQ||EF.   
 Sol. (i) False
(ii) False

3. (i) In Fig. 4.76, DE||AC and DF||AE. Prove that BF/FE=BE/EC .
(ii) Diagonals of a trapezium PQRS intersect each other at the point O, PQ||RS and PQ = 3RS. Find the ratio of the areas of triangles POQ and ROS. 

  Sol. (ii) 9 : 1

Assignments For Class 10 Mathematics Triangles

Question. In the given figure PQ ll BA; PR ll CA.     

Assignments For Class 10 Mathematics Triangles

Solution. In ΔBRD,     
BR ll PQ
Therefore,by basic proportionality theorm,

Assignments For Class 10 Mathematics Triangles

Question. Prove that the area of the equilateral triangle described on the side of an isosceles right angled triangle is half the area of the equilateral triangle described on its hypotenuse.
Solution. Given: In ΔABC in ∠ABC = 90° which and AB = BC.     
Also, ΔABD and ΔACE are equilateral triangles.

Assignments For Class 10 Mathematics Triangles

Since,the ratio of the area of two similar triangles is equal to the square of ratio of their corresponding sides.

Assignments For Class 10 Mathematics Triangles

Question. In an isosceles triangle ABC, if AB =AC= 13 cm and the altitude from A on BC is 5 cm, find BC.
Solution. In ΔADB, by pythagoras theorem,   

Assignments For Class 10 Mathematics Triangles

AD2 + BD2 = AB2
⇒ 52 + BD2 = 132
⇒ 25 + BD2 = 169
⇒ BD2 = 169 – 25 = 144
⇒ BD = √144= 12 cm
In ΔADB and ΔADC
∠ADB = ∠ADC [Each 90°]
AB = AC [Each 13 cm]
AD = AD [Common]
Then, ΔADB ≅ ΔADC [By RHS condition]
∴ BD = CD = 12 cm [By c.p.c.t]
Hence, BC = 12 + 12 = 24 cm.

Question. Let ΔABC ∼ ΔDEF. If ar ( ABC) = 100 cm , ar ( DEF) = 196 cm2 and DE = 7, then find AB.   
Solution. 

Assignments For Class 10 Mathematics Triangles
Assignments For Class 10 Mathematics Triangles

Question. In a trapezium ABCD, diagonals AC and BD intersect at O. If AB = 3CD, then find ratio of areas of triangles COD and AOB. 
Solution. In Δ A O B and Δ C O D   

Assignments For Class 10 Mathematics Triangles

Question. ABC is an isosceles triangle with AB = AC and D is a point on AC such that BC2 = AC x CD. To prove: BD = BC.
Solution. To prove: BD = BC   
proof:BC2 = AC x CD

Assignments For Class 10 Mathematics Triangles

Question. In an equilateral triangle ABC, D is a point on the side BC such the BD = 1/3 BC . Prove that 9AD2 = 7AB2 . 

Assignments For Class 10 Mathematics Triangles

Solution. Given in ΔABC which AB = BC = CA and BD = 1/3BC .   
Construction : Draw AP ⊥BC 

Assignments For Class 10 Mathematics Triangles

Question. 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.
Solution. We have,   

Assignments For Class 10 Mathematics Triangles

In ΔBAD, by mid-point theorem
PS||AD and PS = 1/2 AD ….(i)
In ΔCAD, by mid-point theorem
QR||AD and QR = 1/2 AD ….(ii)
Compare (i) and (ii)
PS || QE and PS = QR
Since one pair of opposite sides is equal as well as parallel then, PQRS is a parallelogram ….(iii)
Now, In ΔABC, by mid-point theorem,
PQ||BC and PQ = 1/2 BC ….(iv)
and, AD = BC …(v) [Given]
Compare equations (i)(iv) and (v)
PS = PQ ….(vi)
From (iii) and (vi),
Since, PQRS is a parallelogram with PS = PQ then PQRS is a rhombus.

Question. State whether the pairs of triangles in the figure are similar or not. Write the similarity criterion used for answering the question and also write the pairs of similar triangles in the symbolic form.   

Assignments For Class 10 Mathematics Triangles

Solution. 

Assignments For Class 10 Mathematics Triangles

Question. In Fig. DEFG is a square and ∠BAC = 90°. Prove that ΔAGF ∼ ΔEFC .   

Assignments For Class 10 Mathematics Triangles

Solution. In ΔAGF and ΔEFC, we have   

Assignments For Class 10 Mathematics Triangles

Question. In Fig. DE ll BC . Then, find AC.   

Assignments For Class 10 Mathematics Triangles

Solution. From figure,   

Assignments For Class 10 Mathematics Triangles

Question. If ΔABC and ΔDEF are triangles such that AB/DE = BC/EF = AC/DF = 4/7 . Find areaΔABC/areaΔDEF .
Solution. 

Assignments For Class 10 Mathematics Triangles

Since for similar triangles,the ratio of the areas is the square of their corresponding sides.   

Assignments For Class 10 Mathematics Triangles

Question. If ∠P = ∠RTS , Then show that ∠PQR = ∠RTS .   

Assignments For Class 10 Mathematics Triangles

Solution. In RPQ and RTS

Assignments For Class 10 Mathematics Triangles
Assignments For Class 10 Mathematics Triangles

Question. Two triangles DEF and GHK are such that D = 48° and H = 57°. If ΔDEF ~ ΔGHK then find the measure of ∠F .
Solution. Given that ΔDEF ΔGHK.
∠D = G=48o …..(Given)
∠E = H=57o …..(Given)
In ΔDEF,
∠D+ ∠E+ ∠F=180o …(Angle sum property)
⇒ 48o +57o+ ∠F=180o
⇒ ∠ F=75o

Question. In Fig. ∠BAC = 90° , AD is its bisector. If DE ⊥ AC , prove that DE x (AB + AC) = AB + AC .
Solution. To prove the given result,we will use the following theorm.
The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle Since AD is the bisector of ∠A of ΔABC. 

Assignments For Class 10 Mathematics Triangles
Assignments For Class 10 Mathematics Triangles

Question. If the diagonals of a quadrilateral divide each other proportionally, then it is a trapezium.   
OR
The diagonals of a quadrilateral ABCD intersect each other at the point O such that AO/BO = CO/DO Show that ABCD is a trapezium.

Assignments For Class 10 Mathematics Triangles

Solution. Through O, draw OE parallel to AB intersecting BC at E.
Now, in ΔABC, OE || AB (Δ by construction)

Assignments For Class 10 Mathematics Triangles
Assignments For Class 10 Mathematics Triangles

Question. In the given figure, P is the mid-point of BC and Q is the mid point of AP. If BQ when produced meets AC at R, prove that RA = 1/3 CA.
Solution. Draw PS || BR, meeting AC at S.   

Assignments For Class 10 Mathematics Triangles

In ΔBCR, P is the mid-point of BC and PS || BR.
∴ S is the mid-point of CR.
⇒ CS = SR …(1)
In ΔAPS, Q is the mid-point of AP and QR || PS.
∴ R is the mid-point of AS.
⇒ AR = RS
from (1) and (2), we get
AR = RS = SC
Now, AC = AR + RS + SC= 3AR
⇒ AR= 1/3 CA Hence proved.

Question. In the given figure, altitudes AD and CE of ΔABC intersect each other at the point P. Show that: 
(i) ΔAEP ~ ΔCDP (ii) ΔABD ~ ΔCBE
(iii) ΔAEP ~ ΔADB (iv) ΔPDC ~ ΔBEC
Solution. (i) In ΔAEP and ΔCDP, we have
∠AEP = ∠CDP (each = 90°)
∠APE = ∠CPD (vertically opposite angles)

Assignments For Class 10 Mathematics Triangles

⇒ ∠AEP ~ ∠CDP (AA similarity criterion)
(ii) In ΔABD and ΔCBE, we have
∠ADB = ∠CEB (each = 90°)
∠ABD = ∠CBE (common)
⇒ ∠ABD ~ ∠CBE (AA similarity criterion)
(iii) In ΔAEP and ΔADB, we have
∠A = ∠A (common)
∠AEP = ∠ADB (each = 90°)
⇒ ∠AEP ~ ∠ADB (AA similarity criterion)
(iv) In ΔPDC and ΔBEC, we have
∠PCD = ∠BCE (common)
∠CDP = ∠CEB (each = 90°)
⇒∠PDC ~ ∠BEC (AA similarity criterion)

Question. In the given figure, DE || OQ and DF || OR. Prove that EF || QR.   
Solution. In ΔPOQ, DE || OR, by Thales’ theorem,

Assignments For Class 10 Mathematics Triangles
Assignments For Class 10 Mathematics Triangles

Question. In an equilateral triangle ABC, D is a point on side BC such that BD= 1/3 BC. Prove that 9AD2 = 7AB2
Solution. Draw AP ⊥ BC.

Assignments For Class 10 Mathematics Triangles
Assignments For Class 10 Mathematics Triangles

Question. ABC is a right triangle right-angled at C. Let BC = a, CA = b, AB = c and let p be the length of perpendicular from C on AB, prove that :

Assignments For Class 10 Mathematics Triangles

Solution.

Assignments For Class 10 Mathematics Triangles

Question. Prove that the diagonals of a trapezium divide each other proportionally. 
OR 
ABCD is a trapezium such that AB || DC. The diagonals AC and BD intersect at O. Prove that AO/BO = CO/DO
Solution. Through O, draw OE || CD (or AB). In ΔABD, EO || AB, by Thales’ theorem,

Assignments For Class 10 Mathematics Triangles

Question. In the given figure, 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
(ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF
Solution. (i) In right ΔOFA, ΔODB and ΔOEC,     
using Pythagoras theorem, we have

Assignments For Class 10 Mathematics Triangles

OA2 = AF2 + OF2
OB2 = BD2 + OD2
OC2 = CE2 + OE2
Adding (1), (2) and (3), we get
OA2 + OB2 + OC2 = AF2 + BD2 + CE2 + OF2 + OD2 + OE2
⇒ OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2
which proves the (i) part.
(ii) In right ΔODB and ΔODC, we have
OB2 = OD2 + BD2
and OC2 = OD2 + CD2
⇒ OB2 – OC2 = BD2 – CD2 …(4)
Similarly, we have
OC2 – OA2 = CE2 – AE2 …(5)
and OA2 – OB2 = AF2 – BF2 …(6)
Adding (4), (5) and (6), we get
(OB2 – OC2)+ (OC2 – OA2) + (OA2 – OB2)= (BD2 – CD2) + (CE2 – AE2) + (AF2 – BF2)
⇒ 0 = (BD2 + CE2 + AF2) – (AE2 + CD2 + BF2)
⇒ AF2 + BD2 + CE2 = AE2 + CD2 + BF2 , which proves the (ii) part.

Question. CD and GH (D and H lie on AB and FE) are respectively bisectors of ∠ACB and ∠EGF and ΔABC ~ ΔFEG. Prove that : (i) CD/GH = AG/FG (ii)ΔDCB ~ ΔHGE (iii)ΔDCA ~ ΔHGF.
Solution. (i) Given ΔABC ~ ΔFEG   
⇒ ∠A = ∠F
and ∠C = ∠G …(1)

Assignments For Class 10 Mathematics Triangles
Assignments For Class 10 Mathematics Triangles

Question. In the given figure, OA/OC = OD/OB . Prove that  ∠A =  ∠C and  ∠B =  ∠D
Solution. In ΔAOD and ΔCOB, we have   

Assignments For Class 10 Mathematics Triangles

Question. In the given figure, XY || AC and XY divides triangular region ABC into two parts equal in area.  10
Determine AX/AB .
Solution. We have, XY || AC and ar (ΔBXY) = ar (quad. XYCA)
⇒ ar (ΔABC) = 2.ar (ΔBXY) …(1)
Now, XY || AC and BA is a transversal.
⇒ ∠BXY= ∠BAC …(2)
Thus, In ΔBAC and ΔBXY, we have
∠XBY = ∠ABC (common)
∠BXY = ∠BAC [from (2)]
⇒ ∠BAC ~ ∠BXY (AA similarity criterion)

Assignments For Class 10 Mathematics Triangles
Assignments For Class 10 Mathematics Triangles

Question. 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 hour. How far apart will be the two planes after 1,1/2 hours? 
Solution. Distance travelled by an aeroplane due north in 1,1/2 hours = 1000 X 3/2 km = 1500 km
∴ OA = 1500 km
Distance travelled by an aeroplane due west in 1,1/2 hours = 1200 X 3/2 km = 1800 km
∴ OB = 1800 km

Assignments For Class 10 Mathematics Triangles

Question. Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.   
OR
ABCD is a rhombus. Prove that AB2 + BC2 + CD2 + DA2 = AC2 + BD2
Solution. Let the diagonals AC and BD of rhombus ABCD intersect at O. Since the diagonals of a rhombus
bisect each other at right angle.
∴ ∠AOB = ∠BOC = ∠COD = ∠DOA = 90° and AO = CO, BO = DO.
Since ΔAOB is a right triangle right-angled at O.
⇒ AB2 = OA2 + OB2 (using Pythagoras theorem)

Assignments For Class 10 Mathematics Triangles
Assignments For Class 10 Mathematics Triangles

Question. Two poles of height a metres and b metres are p metres apart. Prove that the height of the point of intersection of the lines joining the top of each pole to the foot of the opposite pole is given by ab/a + b metres.
Solution. Let AB and CD be two poles of height a metres and b metres respectively such that poles are p metres apart. Let the lines AD and BC meet at O such that OE = h metres.

Assignments For Class 10 Mathematics Triangles

Question. Prove that 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 diagonal.
Solution. Let ABCD be the given square. Equilateral triangles ΔBCE and ΔACF are described on side BC and diagonal AC respectively.

Assignments For Class 10 Mathematics Triangles

Since ΔBCE and ΔACF are equilateral they are equiangular and hence ΔBCE ~ ΔACF (AAA similarity criterion)

Assignments For Class 10 Mathematics Triangles

PRACTICE EXERCISE

Question. The perimeters of two similar triangles ABC and PQR are respectively 36 cm and 24 cm. If PQ = 10 cm, find AB.
Solution. AB = 15 cm

Question. D and E are points on the sides AB and AC respectivley of a ΔABC such that DE || BC and divides ΔABC into two parts, equal in area, find BD/AB .
Solution. 2 -√2/2

Question. In the given figure, AQ and PB are perpendiculars to AB. If AO = 10 cm, BO = 6cm and PB = 9 cm, find AQ.

Assignments For Class 10 Mathematics Triangles

Solution. AQ = 15 cm

Question. Use the given figure to find the value of x if :
(i) OA = 4x – 2, OB = 4x + 1, OC = 2x + 2, OD = 3x – 1
(ii) OA = 3x – 1, OB = 2x + 1, OC = 5x – 3, OD = 6x – 5

Assignments For Class 10 Mathematics Triangles

Solution. (i) x = 5 (ii) x = 2 or x = 1/2

Question. In the given figure, ABC is a triangle, right angled at B. If FG || DE || CB and AG = GE = EB, find DE + FG.

Assignments For Class 10 Mathematics Triangles

Solution. 12 cm

Question. Two poles of heights 6 m and 11 m stand on a plane ground. If the distance betwen their feet is 12 m, find the distance between their tops. 
Solution. 13 m

Question. In the given figure, ΔEDC ~ ΔEBA, ΔBEC = 115° and ΔEDC = 70°. Find :

Assignments For Class 10 Mathematics Triangles

(i) ∠DEC
(ii) ∠DCE
(iii) ∠EAB
(iv) ∠AEB
(v) ∠EBA
Solution. (i) 65 (ii) 45° (iii) 45° (iv) 65° (v) 70°

Question. In the given figure, AO/OC = BO/OD = 1/2 and AB = 5 cm. Find DC.

Assignments For Class 10 Mathematics Triangles

Solution. 10 cm

Question. In ΔABC, DE is parallel to base BC, with D on AB and E on AC. If AD/DB = 2/3 , find BC/DE .

Assignments For Class 10 Mathematics Triangles

Solution. 5/2

Question. In the given figure, AB || CD || EF. If AB = 6 cm, CD = x cm, EF = 10 cm, BD = 4 cm and DE = y cm. Find x and y.

Assignments For Class 10 Mathematics Triangles

Solution. x = 3.75 cm, y = 6.67 cm

Question. A ladder 25 m long reaches a window which is 24 m above the ground on side of the street. Keeping the foot at the same point, the ladder is turned to the other side of the street to reach a window 7 m high. Find the width of the street.
Solution. 31 m

Question. In ΔABC, P divides the side AB such that AP : PB = 1 : 3. Q is a point on AC such that PQ || BC. Find the ratio of the areas of ΔAPQ to trapezium BPQC.
Solution. 1 : 3

Question. In the given figure, DE || BC and AD : DB = 5 : 4. Find ar (ΔDFE)/ar (ΔCFB).

Assignments For Class 10 Mathematics Triangles

Solution. 25 : 81

Question. 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.
Solution. 1.6 m

Question. D and E are points on the sides AB and AC respectively of ΔABC such that DE is parallel to BC and AD : DB = 4:5. CD and BE intersect each other at F. Find the ratio of the areas of ΔDEF and ΔBCF.
Solution. 16 : 81

Question. In the given figure, ∠ABC = 90° and BD ⊥ AC. If BD = 8cm and AD = 4 cm, find CD.

Assignments For Class 10 Mathematics Triangles

Solution. CD = 16 cm

Question. 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. 
Solution. 3.5 cm

Question. In the given figure, DE || BC and AD/DB = 3/5 . If AC = 4.8 cm, find AE.

Assignments For Class 10 Mathematics Triangles

Solution. 1.8 cm

Question. In the given figure, D is a point on AB and E is a point on AC of ΔABC such that DE || BC. If AD : DB = 1:2, AB = 18 cm and AC = 30 cm, find the values of AE, EC, AD and DB.

Assignments For Class 10 Mathematics Triangles

Solution. AE = 10 cm, EC = 20 cm, AD = 6 cm and DB = 12 cm

Question. In the given figure, PQ || RS. Find PT and QT.

Assignments For Class 10 Mathematics Triangles

Solution. PT = 4.8 cm, QT = 5.7 cm

Question. 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. 
Solution. 8.8 cm

Question. ABCD is a trapezium in which AB || DC and AB = 2DC. Determine the ratio of the areas of ΔAOB and ΔCOD.
Solution. 4 : 1

Question. In the given figure, find the value of x if DE || AB.

Assignments For Class 10 Mathematics Triangles

Solution. x = 8 or 11

Question. In the trapezium ABCD, AB || CD and AB = 2CD. If the area of ΔAOB = 84 cm2, find the area of ΔCOD. 
Solution. 21 cm2

Question. In the given figure, DE || BC, AD = 2 cm, BD = 2.5 cm, AE = 3.2 cm and DE = 4 cm. Determine AC and BC.

Assignments For Class 10 Mathematics Triangles

Solution. AC = 7.2 cm, BC = 9 cm

Question. A ladder 25 m long reaches a window of a building 20 m above the ground. Determine the distance of the foot of the ladder from the building.
Solution. 15 m

Question. A vertical stick 15 m long casts its shadow 10 m long on the ground. At the same time, the flag pole casts a shadow 60 m long. Find the height of the pole.
Solution. 72 m

Question. The lengths of the diagonals of a rhombus are 24 cm and 10 cm. Find each side of the rhombus.
Solution. 13 cm

Question. In the given figure, find x in terms of a, b and c.

Assignments For Class 10 Mathematics Triangles

Solution. x = ac/b + c 

Question. In the given figure, ABC is a right triangle right-angled at B. AD and CE are the two medians drawn from A and C respectively. If AC = 5 cm and AD = 3√5/2 cm Find the length of CE.

Assignments For Class 10 Mathematics Triangles

Solution. 2√5 cm

Question. In the given figure, 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.

Assignments For Class 10 Mathematics Triangles

Solution. 12√2 cm

Question. In the given figure, DE || BC and DE : BC = 4 : 5. Find ratio of the areas of ΔADE and trapezium BCED.

Assignments For Class 10 Mathematics Triangles

Solution. 16 : 9

Question. In the given figure, AB || QR. Find PB.

Assignments For Class 10 Mathematics Triangles

Solution. PB = 2 cm

Assignments for Class 10 Mathematics Triangles as per CBSE NCERT pattern

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Assignments For Class 10 Mathematics Triangles
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