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NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.5

December 26, 2018 by Dattu

NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.5 are part of NCERT Solutions for Class 10 Maths. Here are we have given Chapter 6 Triangles Class 10 NCERT Solutions Ex 6.5.

  • Triangles Class 10 Ex 6.1
  • Triangles Class 10 Ex 6.2
  • Triangles Class 10 Ex 6.3
  • Triangles Class 10 Ex 6.4
  • Triangles Class 10 Ex 6.6
BoardCBSE
TextbookNCERT
ClassClass 10
SubjectMaths
ChapterChapter 6
Chapter NameTriangles
ExerciseEx 6.5
Number of Questions Solved17
CategoryNCERT Solutions

NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.5

NCERT Solutions for Class 10 Maths

Page No: 150

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:

(i). Given that sides are 7 cm, 24 cm, and 25 cm. Squaring the lengths of these sides we get 49, 576, and 625.
Clearly, 49 + 576 = 625 or 72 + 242 = 252 .
Therefore, given triangle is satisfying Pythagoras theorem. So, it is a right triangle. The longest side in a right angled triangle is the hypotenuse.
Therefore length of hypotenuse of this triangle = 25 cm.

(ii).Given that sides are 3 cm, 8 cm, and 6 cm. Squaring the lengths of these sides we may get 9, 64, and 36. Clearly, sum of squares of lengths of two sides is not equal to square of length of third side. Therefore given triangle is not satisfying Pythagoras theorem. So, it is not a right triangle

(iii).Given that sides are 50 cm, 80 cm, and 100 cm. Squaring the lengths of these sides we may get 2500, 6400, and 10000. Clearly, sum of squares of lengths of two sides is not equal to square of length of third side. Therefore given triangle is not satisfying Pythagoras theorem. So, it is not a right triangle.

(iv).Given that sides are 13 cm, 12 cm, and 5 cm. Squaring the lengths of these sides we may get 169, 144, and 25. Clearly, 144 +25 = 169 Or, 122 + 52 = 132.
Therefore given triangle is satisfying Pythagoras theorem. So, it is a right triangle.
The longest side in a right angled triangle is the hypotenuse.
Therefore length of hypotenuse of this triangle = 13 cm.

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 × MR.

Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 84
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 85

Question 3. 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 Triangles 86

Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 87
iii.     In  ∆DCA & ∆DAB
∠DCA = ∠DAB = 90º
∠CDA = ∠ADB            (common angle)
∠DAC = ∠DBA            (remaining angle)
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 88

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

Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 89

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

Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 90

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

Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 91

Question 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:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 92
In ∆AOB, ∆BOC, ∆COD, ∆AOD
Applying Pythagoras theorem
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 93

Page No: 151

Question 8. O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 94
(i) OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2 ,
(ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2.

Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 95

Question 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:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 96

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:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 97

Question 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 hours?

Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 98
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 99

Question 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:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 100

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:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 101

Question 14. The perpendicular from A on side BC of a Δ ABC intersects BC at D such that DB = 3CD (see Fig. 6.55). Prove that 2AB2 = 2AC2+ BC2.
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 102

Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 103

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

Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 104

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:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 105

Question 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:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles 106

 

We hope the NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.5 help you. If you have any query regarding NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.5, drop a comment below and we will get back to you at the earliest.

Filed Under: CBSE Tagged With: NCERT Solutions for Class 10 Maths

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