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# ICSE Class 10 Mathematics Revision Notes Chapter 22 - Heights and Distances

Last updated date: 17th Sep 2024
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## Revision Notes for ICSE Class 10 Math Chapter 22 - Free PDF Download

Free PDF download of Class 10 Mathematics Chapter 22 - Heights and Distances Revision Notes & Short Key-notes prepared by our expert Math teachers as per ICSE guidelines. To register Maths Tuitions on Vedantu.com to clear your doubts.

Among the main applications of trigonometry is to calculate the distance between two or more locations, the height of an object, or the angle occupied by any object at a given point without having to measure the distance, height, or angle itself which is thoroughly explained to students in this chapter.

Competitive Exams after 12th Science

## Height and Distance

You can describe the directions of the objects by measuring: angle of elevation and angle of depression. An instrument called a theodolite measures the angles of elevation and depression of objects. Using a rotating telescope, a theodolite measures angles using the principles of trigonometry.

### Angle of Elevation

It is an important concept in trigonometry to understand the angle of elevation as it pertains to height and distance. In physical terms, it is the angle between the horizontal plane and an oblique line between the observer and an object above his head. This angle is ultimately formed above the object. As its name implies, the elevation angle is such that it is above the observer's eyes.

Definition - Elevation is the difference between a horizontal line and a line of sight. An angle of elevation is formed when the line of sight is upward from the horizontal line.

Line of Sight - The line which is drawn from the eyes of the observer to the point being viewed on the object is known as the line of sight. The object is kept above the observer's line of sight in this case. Our altitude or distance can be determined easily if we know the elevation angle.

### Angle of Depression

According to the definition of depression, the angle between the horizontal line and the observation of the object from that line is the depression angle. In essence, we use it to get the distance between two objects when we know the angles and the distance from the ground of an object. It's a 90° angle that is formed between the horizontal line and the line of sight when the line of sight is downward.

### The Angle of Depression Formula

Whenever the line of sight is downward, a 90° angle is formed between the horizontal line and the line of sight.

Tan θ = Opposite Side/Adjacent Side

### Trigonometric Ratios

Trigonometric ratios are used to solve height and distance problems. Any two sides of a triangle are related by the trigonometric ratio of the angle. Thee are as follows-

1. sin⁡θ=height/hypotenuse

2. cos⁡θ=distance/hypotenuse

3. tan⁡θ=height/distance

4. cot⁡θ=distance/height

5. sec⁡θ=hypotenuse/distance

6. cos⁡ecθ=hypotenuse/height

## FAQs on ICSE Class 10 Mathematics Revision Notes Chapter 22 - Heights and Distances

1. According to ICSE Class 10 Maths Chapter 22, what is the difference between Angle of Elevation and Angle of Depression?

As you can see, the angle of elevation is opposite the angle of depression. The elevation is defined as the angle between the line of sight and the horizontal line. The angle formed by the line of sight and a horizontal line is known as the angle of elevation. At an angle of depression, however, the line of sight is downwards to the horizontal line. Example- The angle of elevation is the angle between the horizontal line of sight and an object when someone stands and looks up at it. An angle of depression is determined by looking downward at an object while standing.

2. According to ICSE Class 10 Maths Chapter 22, what is the Angle of Elevation Formula?

Since this is the most asked question in the exam and all the numerical are also based on this formula, students always look for the accurate answer.

So, to determine the angle of elevation, you need to know the opposite side, hypotenuse, and adjacent side of the right angle. The formula for the angle of elevation is given by the formula for the distance from the object and the height of the object.

The angle of elevation angle = height/distance from the target

3. Following the concepts of ICSE Class 10 Maths Chapter 22, solve this

Consider a height of 1.5 m for PQ.

A tower with a height of 22 meters is AB.

The horizontal distance between the tower and the observer is QB

, then PQ = MB = 1.5 meter

AB - MB = AM

Therefore AM = 22 - 1.5 = 20.5

In the ∆APM

=> note: tan θ = AM / PM

tan θ = 20.5 / 20.5

tan θ = 1

The solution to this equation is tan-1(1 ).

Therefore, θ = 45°

Accordingly, the top of the tower is 45 degrees above the observer's eye

4. According to ICSE Class 10 Maths Chapter 22, what is the relation between height and distance?

A height measure indicates an object's height in the vertical direction, while a distance measure indicates the distance separating an object from a particular point in the horizontal direction. If we imagine connecting the observation point to the topmost point of the object, we will have a triangle formed by the horizontal line, vertical line, an imaginary line.

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Assume that the observer is located at point C. Line AB shows the height of the object. Line BC indicates the distance between the observer and the object. This line is not necessarily parallel with the ground. AC represents the Line of Sight when the observer is observing the topmost point of an object. Angle α represents the angle of elevation, while Angle β represents the angle of depression.

5. According to ICSE Class 10 Maths Chapter 22, what does height mean? How do you find the height of objects in Maths?

Generally, height refers to altitude or elevation that is measured above a base level. In addition, height refers to any distance measured above a given level (e.g., from the ground to the top of the head). An example would be the height of a tree, a human being, a mountain, or a tower. Mathematically, you can calculate the height of an object by calculating its distance and angle. In this case, distance is the horizontal distance between the objects, and angle is the angle above the horizontal of the top of the objects, which gives height to the objects.