NCERT Solutions for Class 10 Maths Chapter 6 Triangles (Ex 6.5) Exercise 6.5

The Ch 6 Maths Class 10 is about triangles, their similar figures, how these are configured, and application of Thales theorem and so on. Moving on to Exercise 6.5 Class 10, you will come to the section where Pythagoras Theorem and its converse will be taught to you along with its applications in various cases. There are 17 questions in the exercise in total. Each of them needs step by step solution. But before we tell you about why Vedantu will be your best choice of reference learning, let us take a glance at the contents of the Ex 6.5 Maths Class 10. 

Class 10 Maths Chapter 6 Exercise 6.5 is an extension of what you have been learning throughout the chapter. Pythagoras theorem and its converse are the central theme of Ex 6.5 Class 10. The syllabus aims at making you knowledgeable about various characteristics of triangles and also let you utilise trigonometric identities. NCERT Solutions for Class 10 Maths Chapter 6 will help you get a holistic grip on the subject. Become skilled in dealing with triangles and get higher marks in your Board exams

What is Pythagoras Theorem

According to Pythagoras Theorem, a right triangle has the unique characteristic where the square of its hypotenuse is equal to the sum of the squares of the rest of the two sides. For example, if ∆ABC is a right triangle where the AC side is the hypotenuse, then as per the Pythagoras theorem-, 

AC2= AB2 + BC2

You must know this theorem by heart and also practice the method of proofing this theorem as very often students are asked to do so in the exams. 

What is the converse of Pythagoras theorem

The converse of Pythagoras theorem is about the fact that if in a triangle the sum of squares of two sides is equal to the square of one side, then the angle opposite this side must be a right angle. 

Question 1

In the first question of this exercise 6.5 of Chapter 6 Class 10 Maths, you are given the measurements of the sides of triangles. You have to determine by using Pythagoras theorem which of them would be right triangle and also calculate the length of the hypotenuse of those triangles. Those sides which satisfy the Pythagoras triplet will be termed as the right triangles. But to determine that you have to calculate all the given options and reach a conclusion. 

Question 2

In this question of exercise 6.5 in Class 10 Maths Chapter 6, you are given a pictorial diagram of triangle ∆PQR right-angled at P. Another line PM is placed on the side QR perpendicularly. You have to prove that PM2= QM.MR. To answer this again you have to have proper knowledge about Pythagoras theorem and prove the statement by deriving AAA similarity. 

Question 3

In this question, you have to prove three given statements about sides AB, AC and AD from the given diagram of triangle ∆ABD where AC is perpendicular on BD. In the course of deriving the same, you have to come across AA similarity and AAA similarity. In fact, the first two statements once proved will be useful in proving the last statement. Step by step process will lead you to the desired solution. 

Question 4

In this question from Class 10 Maths Chapter 6 Exercise 6.5, you have to prove the statement AB2= 2AC2 an isosceles triangle ∆ABC right angled at C. This question requires the application of Pythagoras theorem in its simplest form. You can derive the answer yourself quite easily. To check your answer and get assured that you follow the right process we have Triangles Class 10 NCERT Solutions for you. 

Question 5

Here again, you have an isosceles triangle ∆ABC with two sides AC and BC equal with each other. You have to prove that the given triangle is a right triangle. Here you have to apply the converse of Pythagoras theorem and verify whether the sides satisfy the theorem. Previous knowledge of Triangle Class 10 will come in handy in answering this question. 

Question 6

Here you are given a diagram of equilateral triangle ∆ABC with sides 2a. From here you have to find out each of the altitudes of the triangle. To solve this, you have to draw an imaginary line AD, perpendicular on the side BC. Now, your earlier lessons from Chapter 6 Maths Class 10 need to be implemented. You have known that the altitude in equilateral triangle bisects the opposite sides. With the application of Pythagoras theorem in addition to this feature of the triangle, you can solve the problem. 

Question 7

Here you have to put to test your knowledge of equilateral triangles and see how in-depth knowledge you possess is. Here you have to put Pythagoras theorem to use for rhombus and prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals. NCERT Solutions for Class 10 Maths Chapter 6 Triangles by Vedantu will help you in getting the appropriate solution. 

Question 8

This question is again based on the basic concepts of triangles and asks you to prove relations based on those concepts. You need to attempt this answer in step by step process. The NCERT Solutions by Vedantu helps you get a grip on the tricks of solving the geometrical problem and you can develop a deep understanding of the methods to solve. 

Question 9

Here you are presented with a practical problem involving your notion and understanding about trigonometric derivations. Apply Pythagoras theorem with trigonometric calculation of height and distance to calculate the answer. The Exercise 6.5 Class 10 NCERT Solution gives you a detailed explanation of the problem with step by step solution for appropriate understanding. 

Question 10

This again tests your ability to apply the core of Pythagoras theorem. Not only knowing the theorem will help you here. You have to get a grip on the concept to provide a perfect application. In this problem, you are presented with a practical situation from where you have to derive the result. NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.5 elaborates the problem with diagrammatic representation for your better understanding of the application. 

Question 11

The association of Pythagoras Theorem and trigonometric calculation is again the central idea of this problem. Here you need to put more effort to understand the problem at hand and this can only be done when you have complete knowledge about the application of both the theorem and the calculation process of height and distance. NCERT Solutions for Class 10 Maths Chapter 6 Triangles by Vedantu come to your rescue and explain the question for you to understand and solve easily. 

Question 12

The application of Pythagoras in trigonometry continues further in this question of Class 10 Maths Chapter 6 Exercise 6.5. The more you have a clear understanding of the theorem and its applicability in the right situations, the easier it will be to interpret the situation. With the help of NCERT Solutions, it is a job easier done than said especially when you appear for the board exams. 

Question 13

The derivation of Pythagoras theorem and the interrelationship of triangles are the main ideas which you have to cover while solving this problem. Opting for efficient learning in the previous chapters and exercises makes the solution convenient for you to understand. Solving it is furthermore easy with help from the Triangles Class 10 NCERT Solutions. 

Question 14

You are here given a figure of triangle ∆ABC where a perpendicular from point A intersects the BC side of the triangle on point D in a certain manner. From here you have to calculate whether the given statement can be proved or not. Here you are to implement Pythagoras theorem in an elaborate manner. The solution will be quite lengthy. But the Exercise 6.5 Class 10 NCERT Solution will help you understand what steps to follow to solve the problem skillfully. 

Question 15

In this question, you are presented with a bit more complicated diagram of an equilateral triangle ∆ABC with various features of the sides and angles. The statement that you have to prove is 9AD2= 7AB2. You will get to know from the NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.5 that you have to draw an imaginary altitude AE from point A on the side BC. Then you have to apply your knowledge about the altitude of the equilateral triangle which bisects the opposite sides. Step by step calculation will make you reach the conclusion. 

Question 16

Towards the end of exercise 6.5 you get to face problems where you continue to prove some given facts. In this question, for example, you have to prove that three times the square of one side is equal to four times the square of one of its altitudes. Short cut techniques mentioned in the NCERT solutions will enable you to understand the trick of putting your hands upon the problem to solve it with ease. 

Question 17

The last question of the Ex 6.5 Maths Class 10 you are presented with multiple options to choose from where you are asked to calculate one particular angle of the triangle with the lengths of its sides given. Here again, you have to find out whether the three sides are satisfying the Pythagoras triplet and conclude on the triangle being a right angle one. With the NCERT solutions by us, you can solve any such numerical as you can interpret the trick behind the solution. 

Why go for Vedantu only

The NCERT Solutions for Class 10 Maths Chapter 6 covers all the exercises in a subtle manner in which not a single one of the problems are left out. Ready solutions of all of the mathematical problems make your learning and revising easier. 

The problems are solved in a step by step process and so can be interpreted conveniently. In some cases, you are also introduced to short cut techniques meant for solving the problems. This enables you to grasp the technique easily and also solve the question in less time. This is extra beneficial for time management during your board exams. 

Solution by expert Maths tutors makes them absolutely error-free and reliable. This makes these online study materials more preferred by the students across the nation. 

Easy process of solving Maths attracts the attention of even those students who are not quite easy with Maths in class 10. 

All the PDF solution of the exercises can be downloaded from the website and the Smartphone App completely free. There is no bar on the number of times of downloads. You can access them as many times as you want or download them to study as per your convenience. 

You can also Download Maths NCERT Solutions Class 10 to help you to revise complete Syllabus and score more marks in your examinations.

Success with online learning by Vedantu

Registering with vedantu will give you the opportunity to attend Free Live Master Class and Free conceptual videos. You can also take online tests and receive feedback from online expert tutors to polish themselves further for exam preparation. Covering all subjects and including all topics of all chapters enables the NCERT Solutions by Vedantu cover all parts of the syllabus. Not a single point of information is left out. 

All the answers are formulated adhering to the NCERT guidelines. Thus your answers will never deviate from the expected answers by the Board examiners. Complete and thorough concentration on the maintenance of the quality of the content of the NCERT Solutions by Vedantu prepares the students to face the competition not only nationwide but also globally. 

All these make Vedantu solutions your standalone option for online learning and reference. Put a step forward with the digital learning process of Vedantu and reach the zenith of success.