NCERT Solutions for class 10 Maths Chapter 6- Triangles Exercise 6.5

The Chapter 6, Triangles, Exercise 6.5 begins from page number 150 of the NCERT textbook. The exercise covers the portions in which theorems related to right angles are present. In other words, we can say that, the exercise contains questions that could be solved using the theorems that deals with right angles.

The subject experts at BYJU’S make sure that the questions present in the NCERT textbooks are solved in the easiest way possible. The NCERT solutions for class 10 Maths can help you not only with you examination preparation, but also, can be used to check if the answers you gave to the questions are correct or not.

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Access other exercise solutions of class 10 Maths Chapter 6- Triangles

Exercise 6.1 Solutions 3 Questions (3 Short Answer Questions)

Exercise 6.2 Solutions 10 Questions (9 Short Answer Questions, 1 Long Answer Question)

Exercise 6.3 Solutions 16 Questions (1 main question with 6 sub-questions, 12 Short Answer Questions, 3 Long Answer Questions)

Exercise 6.4 Solutions 9 Questions (2 Short Answer with Reasoning Questions, 5 Short Answer Questions, 2 Long Answer Questions)

Exercise 6.6 Solutions 10 Questions (5 Short Answer Questions, 5 Long Answer Questions)

Access Answers of Maths NCERT class 10 Chapter 6- Triangles Exercise 6.5

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:

(i) Given, sides of the triangle are 7 cm, 24 cm, and 25 cm.

Squaring the lengths of the sides of the, we will get 49, 576, and 625.

49 + 576 = 625

(7)2 + (24)2 = (25)2

Therefore, the above equation satisfies, Pythagoras theorem. Hence, it is right angled triangle.

Length of Hypotenuse = 25 cm

(ii) Given, sides of the triangle are 3 cm, 8 cm, and 6 cm.

Squaring the lengths of these sides, we will get 9, 64, and 36.

Clearly, 9 + 36 ≠ 64

Or, 32 + 62 ≠ 82

Therefore, the sum of the squares of the lengths of two sides is not equal to the square of the length of the hypotenuse.

Hence, the given triangle does not satisfies Pythagoras theorem.

(iii) Given, sides of triangle’s are 50 cm, 80 cm, and 100 cm.

Squaring the lengths of these sides, we will get 2500, 6400, and 10000.

However, 2500 + 6400 ≠ 10000

Or, 502 + 802 ≠ 1002

As you can see, the sum of the squares of the lengths of two sides is not equal to the square of the length of the third side.

Therefore, the given triangle does not satisfies Pythagoras theorem.

Hence, it is not a right triangle.

(iv) Given, sides are 13 cm, 12 cm, and 5 cm.

Squaring the lengths of these sides, we will get 169, 144, and 25.

Thus, 144 +25 = 169

Or, 122 + 52 = 132

The sides of the given triangle are satisfying Pythagoras theorem.

Therefore, it is a right triangle.

Hence, length of the hypotenuse of this triangle is 13 cm.

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

Solution: Given, ΔPQR is right angled at P is a point on QR such that PM ⊥QR

NCERT Solutions for Class 10 Maths Chapter 6 - 41

.

We have to prove, PM2 = QM × MR

In ΔPQM, by Pythagoras theorem

PQ2 = PM2 + QM2 

Or, PM2 = PQ2 – QM2 ……………………………..(i)

In ΔPMR, by Pythagoras theorem

PR2 = PM2 + MR2 

Or, PM2 = PR2 – MR2 ………………………………………..(ii)

Adding equation, (i) and (ii), we get,

2PM2 = (PQ2 + PM2) – (QM2 + MR2)

          = QR2 – QM2 – MR2        [∴ QR2 = PQ2 + PR2]

          = (QM + MR)2 – QM2 – MR2

          = 2QM × MR

∴ PM2 = QM × MR

3. In Figure, 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

NCERT Solutions for Class 10 Maths Chapter 6 - 42

Solution:

(i) In ΔADB and ΔCAB,

∠DAB = ∠ACB (Each 90°)

∠ABD = ∠CBA (Common angles)

∴ ΔADB ~ ΔCAB [AA similarity criterion]

⇒ AB/CB = BD/AB

⇒ AB2 = CB × BD

(ii) Let ∠CAB = x

In ΔCBA,

∠CBA = 180° – 90° – x

∠CBA = 90° – x

Similarly, in ΔCAD

∠CAD = 90° – ∠CBA

            = 90° – x

∠CDA = 180° – 90° – (90° – x)

∠CDA = x


In ΔCBA and ΔCAD, we have

∠CBA = ∠CAD

∠CAB = ∠CDA

∠ACB = ∠DCA (Each 90°)

∴ ΔCBA ~ ΔCAD [AAA similarity criterion]

⇒ AC/DC = BC/AC

⇒ AC2 =  DC × BC

(iii) In ΔDCA and ΔDAB,

∠DCA = ∠DAB (Each 90°)

∠CDA = ∠ADB (common angles)

∴ ΔDCA ~ ΔDAB [AA similarity criterion]

⇒ DC/DA = DA/DA

⇒ AD2 = BD × CD

4. ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2.

Solution: Given, ΔABC is an isosceles triangle right angled at C.

NCERT Solutions for Class 10 Maths Chapter 6 - 43

In ΔACB, ∠C = 90°

AC = BC (By isosceles triangle property)

AB2 = AC2 + BC2 [By Pythagoras theorem]

         = AC2 + AC2 [Since, AC = BC]

AB2 = 2AC2

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

Solution: Given, ΔABC is an isosceles triangle having AC = BC and AB2 = 2AC2

NCERT Solutions for Class 10 Maths Chapter 6 - 44

In ΔACB,

AC = BC

AB2 = 2AC2 

AB2 = AC+ AC2

        = AC2 + BC[Since, AC = BC]

Hence, by Pythagoras theorem ΔABC is right angle triangle.

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

Solution: Given, ABC is an equilateral triangle of side 2a. 

NCERT Solutions for Class 10 Maths Chapter 6 - 45

Draw, AD ⊥ BC

In ΔADB and ΔADC,

AB = AC

AD = AD

∠ADB = ∠ADC [Both are 90°]

Therefore, ΔADB ≅ ΔADC by RHS congruence.

Hence, BD = DC [by CPCT]

In right angled ΔADB, 

AB2 = AD+ BD2

(2a)2 = ADa

⇒ AD2  = 4a2 – a2

⇒ AD2  = 3a2 

⇒ AD  = √3a 

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

Solution: Given, ABCD is a rhombus whose diagonals AC and BD intersect at O.

NCERT Solutions for Class 10 Maths Chapter 6 - 46

We have to prove, as per the question, 

AB+ BC+ CD2 + AD= AC+ BD2

Since, the diagonals of a rhombus bisect each other at right angles.

Therefore, AO = CO and BO = DO

In ΔAOB,

∠AOB = 90°

AB2 = AO+ BO…………………….. (i) [By Pythagoras theorem]

Similarly, 

AD2 = AO+ DO…………………….. (ii)

DC2 = DO+ CO…………………….. (iii)

BC2 = CO+ BO…………………….. (iv)

Adding equations (i) + (ii) + (iii) + (iv), we get,

AB+ AD+ DC+ BC2  =  2(AO+ BO+ DO+ CO)

                                           = 4AO+ 4BO[Since, AO = CO and BO =DO]

                                           = (2AO)+ (2BO)2 = AC+ BD2

AB+ AD+ DC+ BC2 = AC+ BD2

Hence, proved.

8. In Fig. 6.54, O is a point in the interior of a triangle.

NCERT Solutions for Class 10 Maths Chapter 6 - 47

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.


Solution: Given, in ΔABC, O is a point in the interior of a triangle.

And OD ⊥ BC, OE ⊥ AC and OF ⊥ AB.


Join OA, OB and OC

NCERT Solutions for Class 10 Maths Chapter 6 - 48

(i) By Pythagoras theorem in ΔAOF, we have

OA2 = OF2 + AF2

Similarly, in ΔBOD

OB2 = OD2 + BD2

Similarly, in ΔCOE

OC2 = OE2 + EC2

Adding these equations,

OA2 + OB2 + OC2 = OF2 + AF2 + OD2 + BD2 + OE+ EC2

OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2.


(ii) AF2 + BD2 + EC2 = (OA2 – OE2) + (OC2 – OD2) + (OB2 – OF2)

∴ AF2 + BD2 + CE2 = AE2 + CD2 + BF2.

9. 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.

Solution: Given, a ladder 10 m long reaches a window 8 m above the ground.

NCERT Solutions for Class 10 Maths Chapter 6 - 49

Let BA be the wall and AC be the ladder,

Therefore, by Pythagoras theorem,

AC2 = AB2 + BC2

102 = 82 + BC2

BC= 100 – 64

BC= 36

BC = 6m

Therefore, the distance of the foot of the ladder from the base of the wall is 6 m.

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: Given, 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.

NCERT Solutions for Class 10 Maths Chapter 6 - 50

Let AB be the pole and AC be the wire.

By Pythagoras theorem,

AC2 = AB2 + BC2

242 = 182 + BC2

BC= 576 – 324

BC= 252

BC = 6√7m

Therefore, the distance from the base is 6√7m.

11. An aeroplane leaves an airport and flies due north at a speed of 1,000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1,200 km per hour. How far apart will be the two planes after
NCERT Solutions for Class 10 Maths Chapter 6 - 51  hours?

Solution: Given,

Speed of first aeroplane = 1000 km/hr

Distance covered by first aeroplane flying due north in
NCERT Solutions for Class 10 Maths Chapter 6 - 51  hours (OA) = 100 × 3/2 km = 1500 km

Speed of second aeroplane = 1200 km/hr

Distance covered by second aeroplane flying due west in
NCERT Solutions for Class 10 Maths Chapter 6 - 52  hours (OB) = 1200 × 3/2 km = 1800 km

NCERT Solutions for Class 10 Maths Chapter 6 - 53

In right angle ΔAOB, by Pythagoras Theorem,

AB2 = AO2 + OB2

⇒ AB2 = (1500)2 + (1800)2

⇒ AB = √(2250000 + 3240000)

            = √5490000

⇒ AB = 300√61 km

Hence, the distance between two aeroplanes will be 300√61 km.

12. 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.

Solution: Given, Two poles of heights 6 m and 11 m stand on a plane ground.

And distance between the feet of the poles is 12 m.

NCERT Solutions for Class 10 Maths Chapter 6 - 54

 

Let AB and CD be the poles of height 6m and 11m.

Therefore, CP = 11 – 6 = 5m


From the figure, it can be observed that AP = 12m

By Pythagoras theorem for ΔAPC, we get,

AP2 = PC2 + AC2

(12m)2 + (5m)2 = (AC)2

AC2 = (144+25)m2 = 169 m2

AC = 13m

Therefore, the distance between their tops is 13 m.


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: Given, D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C.

NCERT Solutions for Class 10 Maths Chapter 6 - 55

By Pythagoras theorem in ΔACE, we get

AC2 + CE2 = AE2 ………………………………………….(i)

In ΔBCD, by Pythagoras theorem, we get

BC2 + CD2 = BD2 ………………………………..(ii)

From equations (i) and (ii), we get,

AC2 + CE2 + BC2 + CD2 = AE2 + BD2 …………..(iii)

In ΔCDE, by Pythagoras theorem, we get

DE2 = CD2 + CE2

In ΔABC, by Pythagoras theorem, we get

AB2 = AC2 + CB2

Putting the above two values in equation (iii), we get

DE2 + AB2 = AE2 + BD2.

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

NCERT Solutions for Class 10 Maths Chapter 6 - 56

Solution: Given, the perpendicular from A on side BC of a Δ ABC intersects BC at D such that;

DB = 3CD.

In Δ ABC,

AD ⊥BC and BD = 3CD

In right angle triangle, ADB and ADC, by Pythagoras theorem,

AB2 = AD2 + BD2 ……………………….(i)

AC2 = AD2 + DC2 ……………………………..(ii) 

Subtracting equation (ii) from equation (i), we get

AB2 – AC2 = BD2 – DC2

                   = 9CD2 – CD2  [Since, BD = 3CD]                

                   = 9CD2 = 8(BC/4)[Since, BC = DB + CD = 3CD + CD = 4CD]

Therefore, AB2 – AC2 = BC2/2

⇒ 2(AB2 – AC2) = BC2

⇒ 2AB2 – 2AC2 = BC2

∴ 2AB2 = 2AC2 + BC2.

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

Solution: Given, ABC is an equilateral triangle.

And D is a point on side BC such that BD = 1/3BC

NCERT Solutions for Class 10 Maths Chapter 6 - 57

Let the side of the equilateral triangle be a, and AE be the altitude of ΔABC.

∴ BE = EC = BC/2 = a/2

And, AE = a√3/2

Given, BD = 1/3BC

∴ BD = a/3

DE = BE – BD = a/2 – a/3 = a/6

In ΔADE, by Pythagoras theorem,

AD2 = AE2 + DE

NCERT Solutions for Class 10 Maths Chapter 6 - 58

⇒ 9 AD2 = 7 AB2



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: Given, an equilateral triangle say ABC,

NCERT Solutions for Class 10 Maths Chapter 6 - 59

Let the sides of the equilateral triangle be of length a, and AE be the altitude of ΔABC.

∴ BE = EC = BC/2 = a/2

In ΔABE, by Pythagoras Theorem, we get

AB2 = AE2 + BE2

NCERT Solutions for Class 10 Maths Chapter 6 - 60

4AE2 = 3a2

⇒ 4 × (Square of altitude) = 3 × (Square of one side)

Hence, proved.

17. Tick the correct answer and justify: In ΔABC, AB = 6√3 cm, AC = 12 cm and BC = 6 cm.
The angle B is:
(A) 120°

(B) 60°
(C) 90° 

(D) 45°


Solution: Given, in ΔABC, AB = 6√3 cm, AC = 12 cm and BC = 6 cm.

NCERT Solutions for Class 10 Maths Chapter 6 - 61


We can observe that,

AB2 = 108

AC2 = 144

And, BC2 = 36

AB2 + BC2 = AC2

The given triangle, ΔABC, is satisfying Pythagoras theorem.

Therefore, the triangle is a right triangle, right-angled at B.

∴ ∠B = 90°

Hence, the correct answer is (C).


NCERT Solutions for class 10 Maths Chapter 6- Triangles Exercise 6.5

The Class 10 Maths Chapter 6, Triangles covers a wide range of topics with the six exercises present in the chapter. The Exercise 6.5 deals with the theorems related to Pythagoras Theorem. 17 Questions are present in this exercise, out of which, 15 questions are short answer type questions and 2 questions are long answer type questions.There are three theorems that comes are covered in this exercise.The three theorem that forms the base of the Exercise 6.5 is given below:

  • Theorem 6.7 : If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.
  • Theorem 6.8 : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It is also known as pythagoras Theorem.
  • Theorem 6.9 : In a triangle, if 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.

The best way to get acquainted with a mathematical concept is to solve as many problems as possible from that particular topic. To familiarise with all the concepts of maths present in class 10, practising with the solutions for the questions given in the NCERT textbook is more than enough. These NCERT Solutions will help the students in scoring excellent marks in the examination.

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