Heights and Distances Problems (Restricted Angles) MCQs Quiz | Class 10
This quiz covers Class X Mathematics (Code 041), Unit V: Trigonometry, focusing on Heights and Distances Problems (Restricted Angles) MCQs. It includes problems using angles of 30, 45, and 60 degrees, involving a maximum of two right triangles. Attempt all 10 multiple-choice questions and then submit to see your score. You can also download a PDF of your answers.
Understanding Heights and Distances with Restricted Angles
Heights and Distances is a practical application of trigonometry that allows us to determine the heights of various objects and the distances between them without direct measurement. This is particularly useful in fields like engineering, architecture, navigation, and surveying.
Key Concepts Explained
- Line of Sight: This is an imaginary line drawn from the eye of an observer to the object being viewed.
- Angle of Elevation: When an observer looks upwards at an object, the angle formed between the line of sight and the horizontal line passing through the observer’s eye is called the angle of elevation.
- Angle of Depression: When an observer looks downwards at an object, the angle formed between the line of sight and the horizontal line passing through the observer’s eye is called the angle of depression.
Note: The angle of elevation of an object from a point is always equal to the angle of depression of the point from the object (assuming the same horizontal level).
Trigonometric Ratios for Standard Angles (30, 45, 60 degrees)
Most problems in Class 10 involving heights and distances utilize the trigonometric ratios (sine, cosine, and tangent) of these specific angles:
| Angle (theta) | sin(theta) | cos(theta) | tan(theta) |
|---|---|---|---|
| 30 degrees | 1/2 | square root 3 / 2 | 1 / square root 3 |
| 45 degrees | 1 / square root 2 | 1 / square root 2 | 1 |
| 60 degrees | square root 3 / 2 | 1/2 | square root 3 |
Remembering these values is crucial for quickly solving problems.
Solving Problems with One or Two Right Triangles
The core strategy for solving heights and distances problems is to identify and use right-angled triangles. Often, a problem can be broken down into one or two such triangles.
- Single Right Triangle: For simple problems (e.g., finding the height of a tower given its distance and angle of elevation), a single right-angled triangle is usually formed. You’ll use one of the trigonometric ratios (commonly tangent, as it relates perpendicular and base) to find the unknown side.
- Two Right Triangles: More complex problems (e.g., finding the height of a building with a flagstaff, or an object observed from two different points) involve two right-angled triangles.
- Typically, these triangles share a common side (e.g., the height of an object or the distance from a point).
- You’ll set up two separate trigonometric equations, one for each triangle.
- Solve these equations simultaneously to find the required unknowns.
- Ensure you draw a clear diagram and label all angles and sides to correctly identify relationships.
Quick Revision Checklist
- Always start by drawing a neat, labeled diagram.
- Identify the right-angled triangles in your diagram.
- Clearly mark the known angles and sides, and the unknown quantities you need to find.
- Choose the correct trigonometric ratio (sin, cos, or tan) that connects the known and unknown sides with the given angle.
- If necessary, solve simultaneous equations when two triangles are involved.
- Use the values of sin, cos, tan for 30, 45, and 60 degrees accurately.
- Angles of elevation and depression are always measured from the horizontal.
Extra Practice Questions
Test your understanding with these additional problems:
- A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is 60 degrees. Find the height of the tower.
- An aeroplane is flying at a height of 300 m above the ground. If the angle of depression from the aeroplane of a point on the ground is 30 degrees, find the distance of the point from the aeroplane.
- The angle of elevation of the top of a building from the foot of the tower is 30 degrees and the angle of elevation of the top of the tower from the foot of the building is 60 degrees. If the tower is 50 m high, find the height of the building.
- A 1.6 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60 degrees. After some time, the angle of elevation reduces to 30 degrees. Find the distance traveled by the balloon during the interval.
- A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60 degrees and from the same point the angle of elevation of the top of the pedestal is 45 degrees. Find the height of the pedestal.
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