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

The Central Board of Secondary Education in New Delhi provides secondary education to both public and private schools. Governed and administered by the Indian government, CBSE is a national education system. The National Council of Educational Research and Training was founded in 1961 with the aim of improving the quality of school education in India. These organisations provide state and federal governments with advisory and assistance services. Moreover, they produce and publish educational kits, educational manuals, textbooks, supplementary materials, newsletters, journals, and multimedia products. For promotion to Class 11 and Class 12, a minimum score of 33% must be achieved by students on both the theory and practical examinations.

Students can improve their mathematical skills, as well as their logical and arithmetic abilities by studying, NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. A simplified explanation of solutions is provided for difficult problems in order to aid in the learning process. In order to prepare for the Class 10 Mathematics examination, students should practise the solutions to the Class 10 Mathematics textbook. It is the purpose of the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 to provide students with answers to all questions in the textbook to assist them in better comprehending the questions. There are many competitive examinations and other college entrance tests that are based on textbooks.Students are advised to regularly practise these NCERT Solutions for Class 10 Maths Chapter 6 Exercises 6.5 in order to achieve higher test scores. On the Extramarks website, students are also encouraged to solve the exemplar problems and provide solutions. It is with the assistance of these solutions that students are able to gain a better understanding of the significance of studying Class 10 Mathematics. Students can access a variety of study materials provided by Extramarks in order to become familiar with the concepts covered in this chapter.

Extramarks provides CBSE Exam Preparation Materials in addition to NCERT Solutions for Class 10 Maths, Chapter 6, Exercise 6.5.It is recommended that students thoroughly practise NCERTSolutions For Class 10 Maths Chapter 6 Exercise 6.5 before taking the final exam. In addition to these solutions, Extramarks also provides solutions for other subjects, their chapters, and even for other classes. Using Extramarks’ website, students in grades 1-12 can easily access these solutions. Experts have curated solutions for all subjects, and they are considered invaluable study resources. For various classes and their respective subjects, students can access the solutions named  NCERT Solutions Class 1, NCERT Solutions Class 2, NCERT Solutions Class 3, NCERT Solutions Class 4, NCERT Solutions Class 5, NCERT Solutions Class 6, NCERT Solutions Class 7, NCERT Solutions Class 8, NCERT Solutions Class 9, NCERT Solutions Class 11, and NCERT Solutions Class 12 on the Extramarks’ website and their mobile application.

It is important for the students to study diligently so that they can gain a thorough understanding of the study material. The learners can improve their answering skills by using the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. Practising these solutions allows students to become familiar with the way they should respond to questions during examinations. Students should be aware of both the fundamentals and the more advanced aspects of Class 10 Mathematics Chapter 6 in order to gain a thorough understanding of the course. For the preparation of final exams and competitive examinations, students are advised to practise these NCERT Solutions.  It is difficult for students to locate adequate authentic NCERT Solutions for Class 10 Maths, Chapter 6, Exercise 6.5.These solutions are easily accessible on the Extramarks website. The purpose of this is to provide students with assistance so that they may prepare effectively for their examinations. On Extramarks’ website and  mobile application, NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 are available in PDF format. It is the ultimate goal to assist students in improving fundamental concepts and gaining a better understanding of Chapter 6 of Class 10 Mathematics. In order to assist students in helping and solving complex problems, NCERT Solutions use a variety of numerical and theoretical concepts.

Following the instructions provided, students acquire a thorough understanding of the key aspects of the various concepts involved. Students may benefit from the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 in order to better understand the subject. In Chapter 6, students can review some important topics using the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. Chapter 6 should be thoroughly studied by students in order to gain a greater understanding of the topics covered. In order to achieve good results, students should thoroughly review the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. Students can gain a thorough understanding of the fundamentals and core concepts of Triangles in this chapter by using the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. Extramarks provides students with access to these NCERT solutions through its website and mobile application. Due to the fact that these solutions have been developed by subject experts, they provide accurate and reliable information. The solutions are developed based on the NCERT textbooks. Additionally, the website provides comprehensive resources and expert-curated NCERT solutions that can assist students in gaining a thorough understanding of the concepts prior to the exam. Extramarks’ NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 are a great study resource for students who are having difficulty understanding the concepts in this chapter.

The solutions to the textbook questions provide a thorough explanation of each question. NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.5 may be useful as students prepare for exams and homework assignments.There is consistent use of simple, straightforward language throughout the PDF document that provides solutions to the problems. All questions are answered in a detailed and comprehensive manner. It is extremely useful for students preparing for competitive exams or assignments to have access to the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. The PDF of the NCERT Solutions is available on Extramarks. The solutions should be reviewed by students as part of their final exam preparation.

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

Students can gain a greater understanding of the topics and concepts covered in Class 10 Mathematics Chapter 6 through NCERT Solutions. Chapter 6 NCERT Textbook questions from the Class 10 textbook should be practised to ensure that students are familiar with the topics covered in the chapter. In order to overcome any difficulties they may encounter when studying Chapter 6 for examinations, students can take assistance from the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. It is recommended that students follow the steps outlined in NCERT Solutions in order to improve their exam performance. The NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 covers all questions from the CBSE textbook. A step-by-step explanation of each question can be found throughout these solutions.

It is recommended that students solve questions from Mathematics Chapter 6 of the NCERT textbook for Class 10 in order to gain a thorough understanding of the concepts presented. The NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 in PDF format can be helpful to students in understanding the subject better. An individual should be able to comprehend both the basic and advanced sections of Chapter 6 in order to gain a thorough understanding of the topic. Prior to taking the final exam, students should practise the NCERT Solutions Class 10 for Mathematics. It also allows them to confidently answer even the most difficult questions during an examination. The Extramarks’ NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 acts as an excellent study guide for students seeking to improve their scores on the CBSE Class 10 Mathematics Board Exam.

Students preparing for competitive exams should make use of Extramarks’ NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6 in PDF format. Extramarks provides detailed solutions to the NCERT textbook problems in accordance with CBSE guidelines. Consequently, students are able to effectively prepare for all topics covered in Mathematics Chapter 6 throughout the academic year. Students may find the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 useful in preparing for competitive exams as well as other exams, such as college entrance exams. An overview of the key formulas is presented at the end of each chapter, followed by examples that are well explained. The NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 are available for students seeking answers to questions in NCERT textbooks. Students in Class 10 use these solutions as study materials, which have been developed after extensive research. It is valuable to have access to NCERT Solutions online.

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

Triangles are the subject of Chapter 6 of the Mathematics syllabus for Class 10. The subject of triangles is one of the most important chapters in Class 10 and is quite useful throughout a students’ academic career in various fields. A discussion of the important aspects of triangles is provided in this chapter, which is complemented by expert-curated solutions. These NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 assist students in gaining a good understanding of the various types of problems that they are going to encounter during their examinations. There is also a discussion of the important properties of triangles in the solutions.

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Triangles – Important Properties

It is important for students in Class 10 to understand the concept of Triangles. In Chapter 6 of the NCERT Textbook for Class 10, various topics are discussed. The fact that all the students have already studied this chapter in their previous classes makes it easier for them to become familiar with it. Triangles, as they are aware, have a number of distinguishing characteristics that set them apart from one another.Throughout the chapter, all of these properties, including sides, angles, and much more, are discussed in detail. In preparation for their Class 10 Board Examination, students are advised to study all the properties carefully.

Students often have difficulty answering the questions related to Chapter 6 Exercise 6.5. Therefore, Extramarks’ website provides access to the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. It is important that students pay attention to the questions in Class 10 Maths Chapter 6 Exercise 6.5 as they prepare for the Class 11 board examination. Students can access a wide range of study materials through Extramarks’ website and mobile application. They can find a variety of resources available on the website to assist them in their preparation, such as past years’ papers, revision notes, sample papers, and some extra questions. Using the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 available on Extramarks, students can prepare for their exams.

To assist students with their studies, experts prepare solutions based on the most recent CBSE syllabus. Students may find these solutions useful in understanding the questions. Students are provided with both solutions and explanations in order to facilitate their understanding of the material. In order to achieve high scores, students can download a PDF version of NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 from Extramarks.

Access NCERT Solutions for Class 10 Maths Chapter – 6 – Triangles

For answers to questions in the Class 11 Mathematics textbook, students should refer to the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. Students should be able to understand the concepts presented in this chapter in depth. Since NCERT books frequently contain examination questions, practising exercises from them is generally beneficial. This chapter contains a number of questions that may be challenging for students to answer. The NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 are available on Extramarks. In order to prepare for the exam, it is recommended that students use authentic and original solutions. Extramarks’ learning platform provides access to reliable study materials, such as NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. Students can accelerate their learning process and improve their academic performance by using these NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. Their understanding of the problems is enhanced by these solutions. The purpose of Extramarks is to provide students with the opportunity to practise solving problems with modules in order to enhance their understanding of the concepts. For students preparing for exams or working on homework assignments, NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 is a valuable resource.

CBSE Class 10 Maths Chapter 6 Exercise 6.5 Solutions

For passing the Class 10 board exams, it is imperative to have a solid understanding of concepts and numericals. To prepare for exams, it is highly recommended that the students use the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. Studying the Extramarks solutions enables students to gain a better understanding of any problem. Through the use of these NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5, this chapter provides comprehensive solutions to all problems posed in Chapter 6. For students to gain a better understanding of the concepts, Extramarks provides solved examples at the end of each chapter. Students can practise and revise concepts more effectively by using NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. A logical approach is required by students in order to perform well on the NCERT Class 10 Mathematics Board Examination. For a quick review of the materials, it is very helpful to have access to NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 readily available when studying. In order to gain a thorough understanding of Chapter 16, students may benefit from reviewing NCERT Solutions For Class 10 Maths Ch 6 Ex 6.5.

NCERT Solutions for Class 10 Maths

To facilitate concept-based learning, NCERT Solutions For Class 10 Mathematics Chapter 6 provides detailed explanations of all questions. As a result of the simple language used in NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5, the questions are more easily retained and understood. Students are also able to track their progress by completing additional practise questions in addition to those provided for each subject. In accordance with the syllabus and examination format, Extramarks provides detailed responses from subject experts. Extramarks provides chapter-by-chapter solutions to all the subjects in Class 10.

It is possible for students to review various chapter-wise pages available on Extramarks’ website in PDF format. Preparing effectively for an examination requires a thorough review of the material prior to the examination.

NCERT Solutions for Class 10 Maths Chapter 6 Exercises

In order to prepare for the examination, students should become familiar with the syllabus. There are a total of six exercises in Chapter 6 of the textbook. The examination includes both short- and long-answer questions.The following table can be used by students to review the same:

Class 10 Maths Exercise 6.5 Solutions

The NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 are available at Extramarks. Students are advised to use these solutions to answer all the questions in the NCERT Class 10 Mathematics textbooks. Each question is thoroughly explained in the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. A team of highly qualified professionals curates these solutions to help students understand and comprehend them thoroughly. This section presents solutions in a step-by-step format that is easy to understand. For students to ensure that they understand the questions, NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 are available in PDF format. TAll questions from the textbook are answered in these NCERT Solutions for Maths Class 10 Chapter 6 Exercise 6.5.The students in Class 10 should refer to the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5.

Exercise 6.5 Class 10 Question 1

As part of Ex 6.5 Class 10, the first question asks the student to measure the sides of four triangles. Pythagoras’ theorem must be applied to each of them, and for further assistance, students can refer to the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5.

Exercise 6.5 Class 10 Question 2

The question 2 of 6.5 Exercise Class 10 presents a pictorial representation of a triangle PQR. By using the similarity criterion and Pythagoras theorem, students can try to find the answer. Still, if they face a problem, they can take assistance from the various study materials provided to them.

Exercise 6.5 Class 10 Question 3

The third question in Mathematics Class 10, Exercise 6.5, presents a triangle ABC with three statements regarding its sides. The students are required to use various properties of triangles, which they are going to study in the chapter.

Exercise 6.5 Class 10 Question 4

The third question in Mathematics Class 10, 6.5, presents a triangle ABC with three statements regarding its sides. These statements must be supported by the triangle properties of similarity.

Exercise 6.5 Class 10 Question 5

In order to prove that it is a right-angle triangle, the converse of Pythagoras theorem must be applied. To answer this question, students must apply the knowledge they gained in class 10 about triangles.

Exercise 6.5 Class 10 Question 6

An ABC triangle is presented in a diagram. The students are required to determine the length of each altitude. To answer this question, they need to use the Pythagorean theorem in addition to this feature of an equilateral triangle.

Exercise 6.5 Class 10 Question 7

In this case, the students need to demonstrate that the square of the sides of a rhombus is equal to the square of the diagonals of the rhombus. To do so, they mustapply the property of equilateral triangles and the Pythagoras theorem.

Exercise 6.5 Class 10 Question 8

Question 8 in exercise 6.5 of Class 10 Mathematics requires the use of the basic concepts of Triangles in order to prove two relations. By approaching this problem in a step-by-step manner, the students should be able to accomplish it easily.

Exercise 6.5 Class 10 Question 9

The problem involves a ladder, and there is a need to determine the distance between two objects. Using the Pythagoras theorem, the students must calculate the height and distance using Trigonometric Derivatives.

Exercise 6.5 Class 10 Question 10

This question can be answered if the students have a solid understanding of Pythagoras theorem. Since it is a practical problem, simply knowing the theorem is not sufficient. The NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 assist them in understanding the derivation.

Exercise 6.5 Class 10 Question 11

In question 10, the students are required to combine their knowledge of Trigonometry and Pythagoras’ theorem to determine the distance between the two planes.

Exercise 6.5 Class 10 Question 12

The Pythagoras theorem and trigonometry concepts are both applied in this problem. Practising such problems and gaining a solid foundation in both of these aspects of Mathematics makes them more comfortable with the concepts during the examination.

Exercise 6.5 Class 10 Question 13

In question thirteen of Exercise 6.5, there are equations involving the Pythagorastheorem and the interrelationship between triangles that must be proven using the sides of a right-angled triangle.

Exercise 6.5 Class 10 Question 14

In this question, the students are given the figure of a triangle ABC, and they must implement Pythagoras’ theorem in detail to get an accurate answer.

Exercise 6.5 Class 10 Question 15

In question 15 of this exercise, the students are given various features of an equilateral triangle. In spite of the fact that it is a complicated problem, once they have gone through the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5, they can easily understand it.

Exercise 6.5 Class 10 Question 16

In order to answer this question, the students must be familiar with some facts about an equilateral triangle. The questions can be easily solved if the students understand the concepts thoroughly.

Exercise 6.5 Class 10 Question 17

Last but not least, there are multiple-choice questions in which the students have to determine the angle of a triangle. The students must study diligently to solve these questions.

Key Features of NCERT Solutions for Class 10 Maths Chapter 6 Exercise 6.5

With the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5, students can study the entire chapter at their own pace. By practising the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5, students can gain an understanding of the importance of studying Class 10 Mathematics Chapter 6 Exercise 6.5. The NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 can be used to study for a variety of competitive exams. Extramarks provides a variety of study materials to assist students in developing an in-depth understanding of the concepts presented in this chapter. Furthermore, the study materials include NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 as well as all the key concepts outlined in the CBSE-recommended syllabus. The solutions, developed by qualified experts, are intended to assist students in better preparing for CBSE exams. The NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 can be used to study for other entrance exams in addition to competitive exams. Prior to the Class 10 board examination, students should practise questions on the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 at least twice. NCERT Solutions are readily available for each chapter to assist students in gaining a thorough understanding of the subject.

Using NCERT textbooks and solutions, students can gain a thorough understanding of each concept. Using these NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5, students can understand all concepts covered in the textbook, including advanced concepts. Extramarks provides students with easy access to the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. These NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 are designed to meet the needs of students studying commerce as a major field of expertise in schools. The NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 are useful because they are written in simple language. One of the most significant advantages of these NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 is that they are available for download both online and offline. This feature is especially useful for students who do not have constant access to the internet. Students studying NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 may find these resources useful. The NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 can be used to prepare for exams and revise for finals.

All the topics covered in the CBSE Class 10 Mathematics Syllabus are thoroughly covered in the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. All major concepts, terminologies, questions, and formulas are explained to students so that they are able to comprehend the material thoroughly. The NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 can help students improve their performance on competitive examinations, aptitude tests, and college entrance exams. To help students gain a comprehensive understanding of the concepts, Extramarks offers NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5. There are detailed explanations of the problems in Class 10 NCERT Solutions. Extramarks’ website provides students with revision notes, sample problems, past years’ papers, assignments, additional questions, worksheets, and, most importantly, NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 to prepare well for the board examination.

The NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 are created by subject experts in consideration of the understanding level of Class 10 students. This makes it easier for students to comprehend them. These NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5 offer the following benefits:

  • By following the NCERT Solutions For Class 10 Maths Chapter 6 Exercise 6.5, the students can clarify their doubts at a deeper level so that they can successfully attempt similar problems independently.
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  • In order to provide the learners with high-quality solutions, highly qualified professionals have taken into account the CBSE pattern as well as the NCERT guidelines.
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Q.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

Ans.
(i) Given sides of the triangle are 7 cm, 24 cm and 25 cm.
Squares of the given sides of the triangle are 49 cm2, 576 cm2 and 625 cm2.
Now,
49 cm2 + 576 cm2 = 625 cm2
Therefore, by converse of Pythagoras theorem, the given triangle is a right triangle.
Also, we know that hypotenuse is the longest side in a right triangle. Thus, length of the hypotenuse is 25 cm.

(ii)

Given sides of the triangle are 3 cm, 8 cm and 6 cm.Squares of the given sides of the triangle are 9 cm2, 64 cm2and 36 cm2.Now,9+36≠64We find that sum of the squares of two sides is not equal tothe square of third side. Therefore, triangle of the given sides is not a right triangle.(iii)Given sides of the triangle are 50 cm, 80 cm and 100 cm.Squares of the given sides of the triangle are 2500 cm2,6400 cm2 and 10000 cm2.Now,2500+6400≠10000We find that sum of the squares of two sides is not equal tothe square of third side. Therefore, triangle of the given sides is not a right triangle.

(iv) Given sides of the triangle are 13 cm, 12 cm and 5 cm.
Squares of the given sides of the triangle are 169 cm2, 144 cm2 and 25 cm2.
Now, 144 c m2 + 25 cm2 = 169 cm2
Therefore, by converse of Pythagoras theorem, the given triangle is a right triangle.
Also, we know that hypotenuse is the longest side in a right triangle. Thus, length of the hypotenuse is 13 cm.

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

Ans.

Let ∠MPR=θIn ΔMPR,∠MRP=180°−90°−θ=90°−θSimilarly, in ΔMPQ,∠MPQ=90°−∠MPR=90°−θ∠MQP=180°−90°−(90°−θ)=θNow, in ΔMPR and ΔMQP, we have∠MQP=∠MPR, ∠MPQ=∠MRP and ∠PMQ=∠PMR.Therefore, by AAA similarity critarion, ΔMPR~ΔMQP.Therefore, we have     QMPM=MPMR⇒PM2=QM×MR

Q.3

In the following figure, ABD is a triangle right angled at A and AC ⊥ BD. Show thati AB2 = BC . BDii AC2 = BC . DCiii AD2 = BD . CD

Ans.

(i)In ΔADB and ΔCAB, we have            ∠DAB=∠ACB=90°and    ∠ABD=∠CBA [Common angle]Therefore, by AA similarity critarion, ΔADB~ΔCAB.Therefore, we have     ABCB=BDAB⇒AB2=BC⋅BD(ii)Let ∠CAB=θIn ΔCBA, we have∠CBA=180°−90°−θ=90°−θIn ΔCAD, we have∠CAD=90°−∠CAB=90°−θ∠CDA=180°−90°−(90°−θ)=θIn ΔCBA and ΔCAD, we have∠CBA=∠CAD,  ∠CAB=∠CDA and ∠ACB=∠DCA=90°Therefore, by AAA similarity critarion, ΔCBA~ΔCAD.Therefore, we have     ACDC=BCAC⇒AC2=BC⋅DC(iii)In ΔDCA and ΔDAB, ∠DCA=∠DAB=90°, ∠CDA=∠ADBTherefore, by AA similarity critarion, ΔDCA~ΔDAB.Therefore, we have     DCDA=DADB⇒AD2=BD⋅CD

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

Ans.
Given that ABC is an isosceles triangle right angled at C.
Therefore, AC = BC

Using Pythagoras theorem in the given triangle,
we have
AB2 = AC2 + BC2 = AC2 + AC2 = 2AC2

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

Ans.

Given that ABC is an isosceles triangle with AC = BC and AB2 = 2AC2.
Therefore,
AB2 = 2AC2 = AC2 + BC2
Therefore, by converse of Pythagoras theorem, ABC is a right triangle.

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

Ans.

It is given that ABC is an equilateral triangle of side 2a.Let AD is an altitude.We know that altitude bisects opposite side in anequilateral triangle.Therefore, BD=CD=a.Using Pythagoras theorem in ΔADB, we get      AB2=BD2+AD2⇒AD2=AB2−BD2=(2a)2−a2=4a2−a2=3a2⇒AD=a3We know that all the altitudes in an equilateral triangleare of same length. Therefore, length of each altitude is a3.

Q.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.

Ans.

Let ABCD is a rhombus. We know that diagonals of a rhombus bisect each otherat right angles.Using Pythagoras Theorem in ΔAOB, ΔBOC, ΔCOD and ΔAOD, we getAB2=OA2+OB2, BC2=OB2+OC2,CD2=OC2+OD2,DA2=OA2+OD2Adding all these equations, we getAB2+BC2+CD2+DA2=2(OA2+OB2+OC2+OD2)                                              =22AC22+2BD22                                              =AC2+BD2

Q.8

In the following figure, O is a point in the interior of a triangle ABC,OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. (i) OA2+OB2+OC2− OD2−OE2−OF2 = AF2+BD2+CE2,(ii)  AF2+BD2+CE2 = AE2+CD2+BF2.

Ans.

We join O to A, B and C.(i)In ΔAOF,OA2=OF2+AF2 [By Pythagoras theorem]In ΔBOD,OB2=OD2+BD2 [By Pythagoras theorem]In ΔCOE,OC2=OE2+EC2 [By Pythagoras theorem]Adding these equations, we get OA2+OB2+OC2=OF2+AF2+OD2+BD2+OE2+EC2⇒OA2+OB2+OC2−OD2−OE2−OF2=AF2+BD2+EC2(ii)We have form above result,AF2+BD2+EC2=OA2+OB2+OC2−OD2−OE2−OF2                                 =(OA2−OE2)+(OB2−OF2)+(OC2−OD2)                                 =AE2+BF2+CD2

Q.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.

Ans.

Let AB is a ladder and AC is wall. A represents the windowand BC is the distance of the foot of the ladder from thebase of the wall.ABC is a right angle triangle. Therefore, using Pythagorastheorem, we get     AB2=BC2+AC2⇒ 102=BC2+82⇒BC2=100−64=36⇒BC=6Therefore, the distance of the foot of the ladder from thebase of the wall is 6 m.

Q.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?

Ans.

Let AC is a pole and AB is a guy wire with stake at B. Then, ABC is a right angle triangle. Therefore, using Pythagorastheorem, we get     AB2=BC2+AC2⇒ 242=BC2+182⇒BC2=242−182=576−324=252⇒BC=252=4×9×7=67Therefore, the distance of the foot of the ladder from thebase of the wall is 67 m.

Q.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 ata speed of 1200 km per hour. How far apart will be the two planes after 112  hours?

Ans.

Distance travelled by the plane flying towards north in112 hours = 1000×112 km=1500 kmDistance travelled by the plane flying towards west in112 hours = 1200×112 km=1800 kmWe represent thsese distances by OA and OB.Also, AB represents the distance between the two planes.Using Pythagoras theorem, we get AB2=OA2+OB2=15002+18002=22,50,000+32,40,000⇒AB2=54,90,000⇒AB=54,90,000=9×6,10,000=30061Therefore, distance between the two planes=30061 km

Q.12

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

Ans.

Let AB and CD be two poles of heights 6 m and 11mrespectively. Let their feet are on the ground at points Band D. It is given that distance between feet is 12 m. Therefore,BD=12 m.Also,AB=DE=6mandCE=CD−DE=11 m−6 m=5 mNow, AEC is a right triangle. Therefore, by Pythagoras theorem,AC=AE2+CE2=122+52=144+25=169=13 mTherefore,Distance between tops of the two poles=13 m

Q.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.

Ans.

In ΔACE,       AE2=AC2+CE2                                                                 ...(1)In ΔDCB, BD2=CD2+BC2                                                               ...(2)On adding (1) and (2), we haveAE2+BD2=AC2+CE2+CD2+BC2                                       ...(3)In ΔACB,AB2=AC2+BC2                                                                            ...(4)In ΔDCE,DE2=CD2+CE2                                                                             ...(5)On adding (4) and (5), we haveAB2+DE2=AC2+BC2 +CD2+CE2                                         ...(6)From (3) and (6), we find thatAE2+BD2=AB2+DE2

Q.14

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

Ans.

In ΔADC,       AC2=CD2+AD2⇒AD2= AC2−CD2                                        ...(1)In ΔADB, AB2=AD2+DB2⇒AD2=AB2−DB2                                        ...(2)From (1) and (2), we haveAC2−CD2=AB2−DB2                                          ...(3)Now, it is given that DB=3CDAlso,       BC=CD+DB⇒BC=CD+3CD=4CD⇒CD=BC4                                                                 ...(4)Again,      BC=CD+DB⇒BC=DB3+DB=4DB3⇒DB=3BC4                                                                 ...(5)From (3), (4) and (5), we have       AC2−CD2=AB2−DB2⇒AC2−BC216 = AB2−9BC216⇒16AC2−BC2=16AB2−9BC2⇒16AB2=16AC2−BC2+9BC2=16AC2+8BC2⇒2AB2=2AC2+BC2

Q.15 

In an equilateral triangle ABC, D is a point on side BC such that BD=13BC. Prove that 9AD2=7AB2.

Ans.

It is given that ABC is an equilateral triangle.Let length of each side of ΔABC be a.It is given that D is a point on side BC such that BD=13BC=a3.Let AE is an altitude of equilateral triangle ABC. So,BE=EC=12BC=a2.Also, AE=AC2−EC2=a2−a24=32aAgain, DE=BE−BD=a2−a3=a6In ΔADE,      AD2=AE2+DE2 [By Pythagoras Theorem]⇒AD2=(32a)2+a236=34a2+a236=27a2+a236=2836a2=79a2⇒AD2=79a2=79AB2 [AB=a]⇒9AD2=7AB2

Q.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.

Ans.

Let ABC is an equilateral triangle and AE is its altitude.Let length of each side of ΔABC be a.So,BE=EC=12BC=a2.Using Pythagoras theorem in ΔACE, we have AE2=AC2−EC2=a2−a24=34a2⇒3a2=4AE2Therefore, three times the square of one side of anequilateral triangle is equal to four times the squareof one of its altitude.

Q.17

Tick the correct answer and justify: In Δ ABC, AB=63 cm, AC=12 cm and BC=6 cm.The angle B is :(A) 120°   (B) 60°(C) 90° (D) 45°

Ans.

It is given that in ΔABC, AB=63 cm, AC=12 cm and BC=6 cm⇒AB2=108 cm2, AC2=144 cm2 and BC2=36 cm2Now,AB2+BC2=108 cm2+36 cm2=144 cm2=AC2Therefore, by converse of Pythagoras theorem, we find thatin ΔABC, angle B is 90°.Therefore, correct answer is (C).

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