NCERT Solutions For Class 10 Maths Chapter 6 Triangles

NCERT Solutions For Class 10 Maths Chapter 6 Triangles

NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1

Ex 6.1 Class 10 Maths Question 1.
Fill in the blanks by 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)
Solution:

NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.1 Triangles in English medium PDF

Ex 6.1 Class 10 Maths Question 2.
Give two different examples of pairs of
(i) similar figures.
(ii) non-similar figures.
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.1 updated for 2019-2020

Ex 6.1 Class 10 Maths Question 3.
State whether the following quadrilaterals are similar or not.
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q1
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.1 trinagles in hindi

Triangles Class 10 Ex 6.2

Ex 6.2 Class 10 Maths Question 1.
In the given figure (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q2
Solution:
Triangles Class 10 Exercise 6.2

Ex 6.2 Class 10 Maths Question 2.
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
Solution:
Triangles Class 10 Exercise 6.2

Ex 6.2 Class 10 Maths Question 3.
In the given figure, if LM || CB and LN || CD.
Prove that AM/AB=AN/AD
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q3
Solution:
exercise 6.2 class 10

Ex 6.2 Class 10 Maths Question 4.
In the given figure, DE || AC and DF || AE.
Prove that BF/FE=BE/EC
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q4
Solution:
class 10 maths triangles

Ex 6.2 Class 10 Maths Question 5.
In the given figure, DE || OQ and DF || OR. Show that EF || QR.
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q5
Solution:
Class 10 Triangles Ex 6.2

Ex 6.2 Class 10 Maths Question 6.
In the given figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q6
Solution:
triangles class 10

Ex 6.2 Class 10 Maths Question 7.
Using B.P.T., 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 your have proved it in class IX)
Solution:
ncert solutions for class 10 maths chapter 6

Ex 6.2 Class 10 Maths Question 8.
Using converse of B.P.T., prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side.  (Recall that your have done it in class IX)
Solution:
ch 6 maths class 10

Ex 6.2 Class 10 Maths Question 9.
ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that AO/BO=CO/DO
Solution:
triangles class 10 ncert solutions

Ex 6.2 Class 10 Maths Question 10.
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.
Solution:
similar triangles class 10

Triangles Class 10 Ex 6.3

Ex 6.3 Class 10 Maths Question 1.
State which pairs of triangles in the given 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 :
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q7

NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q8
Solution:
ncert solutions for class 10 maths chapter 6 triangles
ncert solutions for class 10 maths chapter 6 pdf

Ex 6.3 Class 10 Maths Question 2.
In the given figure, ∆ODC ~ ∆OBA, ∠BOC = 125° and ∠CDO = 70°. Find ∠DOC, ∠DCO and ∠OAB.
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q9
Solution:
ncert solutions for class 10 maths triangles

Ex 6.3 Class 10 Maths Question 3.
Diagonals AC and BD of a trape∠ium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that OA/OC=OB/OD
Solution:
triangles class 10 solutions

Ex 6.3 Class 10 Maths Question 4.
In the given figure, QR/QS=QT/PR and ∠1 = ∠2. show that ∆PQR ~ ∆TQR.
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q10
Solution:
similarity of triangles class 10

Ex 6.3 Class 10 Maths Question 5.
S and T are points on sides PR and QR of ∆PQR such that ∠P = ∠RTS. Show that ∆RPQ ~ ∆RTS.
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.3 Triangles in English medium PDF

Ex 6.3 Class 10 Maths Question 6.
In the given figure, if ∆ABE ≅ ∆ACD, show that ∆ADE ~ ∆ABC.
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q11
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.3 Tribhuj in Hindi medium PDF

Ex 6.3 Class 10 Maths Question 7.
In the given figure, 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
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q12
Solution:
Triangles Class 10 Ex 6.3

Ex 6.3 Class 10 Maths Question 8.
E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ∆ABE ~ ∆CFB.
Solution:
Triangles Class 10 Exercise 6.3

Ex 6.3 Class 10 Maths Question 9.
In the given figure, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that:
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q13
Solution:
exercise 6.3 class 10

Ex 6.3 Class 10 Maths Question 10.
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
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q14
Solution:
class 10 maths triangles ex 6.3
exercise 6.3 class 10 ncert solutions

Ex 6.3 Class 10 Maths Question 11.
In the given figure, 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.
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q15
Solution:
Class 10 Triangles Exercise 6.3

Ex 6.3 Class 10 Maths Question 12.
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 in given figure). Show that ∆ABC ~ ∆bPQR.
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q16
Solution:
Chapter 6 Maths Class 10 Ex 6.3 NCERT Solutions PDF

Ex 6.3 Class 10 Maths Question 13.
D is a point on the side BC of a triangle ABC, such that ∠ADC = ∠BAC. Show that CA² = CB.CD.
Solution:
similar triangles class 10 ex 6.3

Ex 6.3 Class 10 Maths Question 14.
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.
Solution:
chapter 6 maths class 10 ex 6.3

Ex 6.3 Class 10 Maths Question 15.
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.
Solution:
class 10 maths chapter 6 ex 6.3

Ex 6.3 Class 10 Maths Question 16.
If AD and PM are medians of triangles ABC and PQR respectively, where
∆ABC ~ ∆PQR. Prove that AB/PQ=AD/PM
Solution:
ncert solutions for class 10 maths chapter 6 triangles ex 6.3

Triangles Class 10 Ex 6.4

Ex 6.4 Class 10 Maths Question 1.
Let ∆ABC ~ ∆DEF and their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC.
Solution:
ncert solutions for class 10 maths chapter 6 ex 6.4 pdf

Ex 6.4 Class 10 Maths Question 2.
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.
Solution:
cbse class 10 maths triangles ex 6.4 ncert solutions

Ex 6.4 Class 10 Maths Question 3.
In the given figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that: ar(ABC)/ar(DBC)=AO/DO
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q17
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.4 updated for 2019-2020

Ex 6.4 Class 10 Maths Question 4.
If the areas of two similar triangles are equal, prove that they are congruent.
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.4 Tribhuj in Hindi medium PDF

Ex 6.4 Class 10 Maths Question 5.
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.
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.4 trinagles in hindi

Ex 6.4 Class 10 Maths Question 6.
Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
Solution:
Triangles Class 10 Ex 6.4

Ex 6.4 Class 10 Maths Question 7.
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.
Solution:
Triangles Class 10 Exercise 6.4

Ex 6.4 Class 10 Maths Question 8.
Tick the correct answer and justify
(i) 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
ex 6.4 class 10
(ii) 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
exercise 6.4 class 10

Triangles Class 10 Ex 6.5

Ex 6.5 Class 10 Maths Question 1.
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) 13 cm, 12 cm, 5 cm
Solution:
solution of class 10 maths ex 6.5

Ex 6.5 Class 10 Maths Question 2.
PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM2 = QM X MR.
Solution:
ncert solutions for class 10 maths chapter 6 triangles ex 6.5

Ex 6.5 Class 10 Maths Question 3.
In the given figure, ABD is a triangle right angled at A and AC i. BD. Show that
(i) AB2 = BC.BD
(ii) AC2 = BC.DC
(iii) AD2 = BD.CD
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q18
Solution:
cbse class 10 maths triangles ex 6.5 ncert solutions

Ex 6.5 Class 10 Maths Question 4.
ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2.
Solution:
ncert solutions for class 10 maths triangles ex 6.5

Ex 6.5 Class 10 Maths Question 5.
ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2 , Prove that ABC is a right triangle.
Solution:
Triangles Class 10 Ex 6.5

Ex 6.5 Class 10 Maths Question 6.
ABC is an equilateral triangle of side la. Find each of its altitudes.
Solution:
Triangles Class 10 Exercise 6.5

Ex 6.5 Class 10 Maths Question 7.
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
Solution:
ex 6.5 class 10

Ex 6.5 Class 10 Maths Question 8.
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 + BF2.
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q19
Solution:
exercise 6.5 class 10 ncert solutions

Ex 6.5 Class 10 Maths Question 9.
A ladder 10 m long reaches a window 8 m above the ground. ind the distance of the foot of the ladder from base of the wall.
Solution:
Class 10 Triangles Ex 6.5

Ex 6.5 Class 10 Maths Question 10.
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?
Solution:
Class 10 Triangles Exercise 6.5

Ex 6.5 Class 10 Maths Question 11.
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:
similar triangles class 10 ex 6.5

Ex 6.5 Class 10 Maths Question 12.
Two poles of heights 6 m and 11m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.
Solution:
chapter 6 maths class 10 ex 6.5

Ex 6.5 Class 10 Maths Question 13.
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.
Solution:
class 10 maths chapter 6 ex 6.5

Ex 6.5 Class 10 Maths Question 14.
The perpendicular from A on side BC of a ∆ABC intersects BC at D such that DB = 3CD (see the figure). Prove that 2AB= 2AC2 + BC2.
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q20
Solution:
Ch 6 Maths Class 10 Ex 6.5 NCERT Solutions PDF Q14

Ex 6.5 Class 10 Maths Question 15.
In an equilateral triangle ABC, D is a point on side BC, such that BD = 1/3BC. Prove that 9AD2 = 7AB2.
Solution:
Ch 6 Maths Class 10 Ex 6.5 NCERT Solutions PDF Q15

Ex 6.5 Class 10 Maths Question 16.
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.
Solution:
Class 10 Maths Chapter 6 Ex 6.5 NCERT Solutions PDF Q16

Ex 6.5 Class 10 Maths Question 17.
Tick the correct answer and justify : In ∆ABC, AB = 6√3cm, AC = 12 cm and BC = 6 cm. The angle B is:
(a) 120°
(b) 60°
(c) 90°
(d) 45
Solution:
Class 10 Maths Chapter 6 Ex 6.5 NCERT Solutions PDF Q17

Triangles Class 10 Ex 6.6

Ex 6.6 Class 10 Maths Question 1.
In the given figure, PS is the bisector of ∠QPR of ∆PQR. Prove that QS/SR=PQ/PR
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q21
Solution:
ncert solutions for class 10 maths chapter 6 ex 6.6 pdf

Ex 6.6 Class 10 Maths Question 2.
In the given figure, D is a point on hypotenuse AC of ∆ABC, DM ⊥ BC and DN ⊥ AB. Prove that:
(i) DM2 = DN X MC
(ii) DN2 = DM X AN
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q22
Solution:
ncert solutions for class 10 maths triangles ex 6.6
triangles ex 6.6 class 10 solutions

Ex 6.6 Class 10 Maths Question 3.
In the given figure, ABc is triangle in which ∠ABC > 90° and AD ⊥ CB produced. Prove that AC2 = AB2 + BC2  + 2BC X BD
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.1 Q23
Solution:
similarity of triangles ex 6.6 class 10

Ex 6.6 Class 10 Maths Question 4.
In the given figure, ABC is atriangle in which ∠ABC 90° and AD ⊥ CB. Prove that AC2 = AB2 + BC2 – 2BC X BD
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.6 Q24
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.6 updated for 2019-2020

Ex 6.6 Class 10 Maths Question 5.
In the given figure, Ad is a median of a triangle ABC and AM ⊥ BC. Prove that
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.6 Q5
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.6 trinagles in hindi
Triangles Class 10 Ex 6.6

Ex 6.6 Class 10 Maths Question 6.
Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
Solution:
Triangles Class 10 Exercise 6.6

Ex 6.6 Class 10 Maths Question 7.
In the given figure, two chords AB and CD intersect each other at the point P. Prove that:
(i) ∆APC ~∆DPB
(ii) AP X PB = CP X DP
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.6 Q7
Solution:
exercise 6.6 class 10

Ex 6.6 Class 10 Maths Question 8.
In the given figure, 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 X PB = PC X PD
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.6 Q8
Solution:
class 10 maths triangles

Ex 6.6 Class 10 Maths Question 9.
In the given figure, D is a point on side BC of ∆ABC, such that BD/CD=AB/AC Prove that AD is the bisector of ∆BAC.
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.6 Q9
Solution:
Class 10 Triangles Ex 6.6

Ex 6.6 Class 10 Maths Question 10.
Nazima is fly fishing in a stream. The trip of her fishing rod is 1.8m above the surface of the water and the fly at the end of the string rests on the water 3.6m away and 2.4 m from a point directly under the trip of the rod. Assuming that her string (from the trip of the rod to the fly) is that, how much string does she have out (see the figure)? If she pills 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?
NCERT Solutions For Class 10 Maths Chapter 6 Triangles Ex 6.6 Q10
Solution:
Chapter 6 Maths Class 10 Ex 6.6 NCERT Solutions PDF Q10

Triangles Class 10 Extra Questions Maths Chapter 6

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