Xam Idea Class 10 Maths Chapter 11 Solutions: Heights And Distances
Xam Idea Class 10 Maths Solutions Chapter 11 ‘Heights and Distances’ have been compiled in order to help you in solving the exercises in this chapter effortlessly. This chapter is mainly about the applications of the concepts of trigonometry explained in the previous chapter Xam Idea Solutions. In this chapter, you will learn about the line of sight between the observer and the object under observation. You will also come across the discussion about how the location of the line of sight can define the angle of elevation or angle of depression besides learning the role of the horizontal level in chapter 11 ‘heights and distances’. Using the trigonometric ratios, you will be able to apply the concepts in finding heights and distances of various objects in daily life.
Chapter 11 ‘Heights and Distances’ in the Xam Idea Class 10 book comprises 38 questions categorized systematically in 6 exercises. These exercises are based on the length of the answers and their weightage in the school exams. You will get to solve questions based on heights and distances in this chapter in different formats. The formats include very short answer type and two sections of short answer type questions, long answer type questions, HOTS (Higher Order Thinking Skills) questions, and value-based questions. Practising these questions will help you outshine your batchmates by a significant margin, be it your school exam or Maths Olympiads.
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Important Topics for Xam Idea Class 10 Maths Solutions Chapter 11: Heights and Distances
- This chapter covers the practical applications of the trigonometric ratios and identities that were discussed in chapter 10 ‘Introduction to Trigonometry’ thoroughly.
- These topics were used in astronomy in ancient times. Trigonometric ratios can also be used in geography and earth sciences besides their applications in navigation for the construction of maps, determination of an island with respect to its given latitudes and longitudes.
- A theodolite is an instrument used for surveying which is used for measuring angles with a rotating telescope by using the principles of trigonometry. A theodolite helps in measuring the angle of depression or angle of elevation which is discussed in the sections ahead.
- In India, two largest-ever theodolites were built under a surveying project, popularly known as ‘Great Trigonometric Survey’. These theodolites are now available for display in the museum of the Survey of India which is located in Dehradun.
- It was in the year 1856 when the large theodolite was used for the first time by Sir George Everest.
- This chapter will cover the methods and steps of evaluating heights and distances of various objects, without having to actually measure them.
Some Important Definitions
We must understand some important terms to build a hypothetical right-angled triangle in practical situations to be able to apply the trigonometric ratios as a problem-solving technique. These terms are the line of sight, which makes an angle with the horizontal level. These angles have also been classified with respect to the horizontal as discussed below:
- Line of Sight: Line of sight is the line segment joining the point where the eye of an observer lies and the point of the object being viewed by the observer. For example, if a student looks at the top of the Qutb Minar, the line joining her eyes to the tip of the Minar will be termed as the line of sight.
- Horizontal Level: This term simply means the horizontal axis that is parallel to the earth’s surface and passes through the eye of the observer. Horizontal plays a crucial role in defining the angle of elevation and angle of depression.
- The angle of Elevation: The angle which is formed between the line of sight and the horizontal when the object lies above the horizontal is known as the Angle of Elevation. We call it the angle of elevation because the observer has to raise her eyes to view the given object.
- The angle of Depression: The angle which is formed between the line of sight and the horizontal when the object lies below the horizontal is known as the Angle of Depression. We term it as the angle of depression because the observer has to lower her eyes to view the given object.
Heights and Distances
The first step to measure the height or distance in any given situation is to mark the line of sight and figure out the value of the angle of elevation or depression.
Once we figure out what is the angle of depression or angle of elevation in a given question, we can apply the trigonometric ratios to find the unknown distance or height of any object such as a TV tower, height of given poles, etc.
If the angle is one of those values mentioned in the table given in the previous chapter, you can directly use the memorized values and save your time. It is of vital importance that we learn to draw the correct diagrams for each word problem. Also, these diagrams will fetch you more marks along with helping you to find an accurate answer.
Points to Remember
Below are some important points that will help you in approaching a word problem:
- The angle of depression or the angle of elevation is always an acute angle (less than 900)
- When solving problems, if the height of the observer is not mentioned, we represent the observer as a point.
- While solving problems, if the height of the object is given, then it is represented by a line segment whereas, in the absence of data, we shall represent the object using a point.
- When solving questions for a shadow, as the sun goes down, the length of the shadow increases and vice versa.
Exercise Discussion of Xam Idea Class 10 Maths Solutions Chapter 11: Heights and Distances
Very Short Answer Type Questions
The answers to the questions in this exercise will require you to present the solution in one or two lines with adequate reasoning. These questions are generally asked in the one-marks section in exams and therefore, can help you score full marks from this topic. These questions are based on finding heights and distances using simple trigonometric ratios.
Short Answer Type Questions
There are two sections of this exercise comprising 8 and 6 questions respectively. These questions are mostly formula based in which you can easily find the required heights or distances by using the trigonometric ratios, either directly or by using the trigonometric ratios for complementary angles.
Long Answer Type Questions
There are 8 questions in this exercise in which diagrams are very important. You will need to find the height and distances in the respective questions through a thorough understanding of their diagrammatical representation. These questions are among the most recurring in the school exams.
Higher Order Thinking Skills
The questions in this chapter include complex word problems such as that of the deck of a ship with respect to a hill, or two poles relative to one another, or distance between a pole and a building while their angles of elevation are given, etc.
This exercise comprises questions which are of practical importance and will help you brush up your concepts at the end of the chapter. The questions in this exercise cover situational questions such as that of a broken tree, a circus artist walking on a rope, or a person standing on the bank of a river, etc.
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