medium difficulty6 min

One-observer height problem

A one-observer height problem is a trigonometry question where one observer, one angle, and one horizontal distance are used to find a height or distance.

In one line: One observer means one triangle, one angle, one main ratio.

Practice This Concept

Understand it

These problems are the standard class 10 application type.
One person stands at a known distance from a tower, tree, or pole and measures the angle of elevation or depression.
Then one right triangle is formed.
The unknown height or horizontal distance is found by choosing the correct ratio, usually tan.
The method is simple: sketch, identify sides, write the ratio, and solve.

Example

A student 15 m away from a building sees its top at 45 degrees. Then tan 45 = height/15, so the building height is 15 m.

Formula and rule

For one observer on level ground, tan θ = height / horizontal distance is often the key relation.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1A person stands 12 m from a pole and sees the top at 60 degrees.
2Using tan 60 = height/12, height = 12 root 3 m.

Method you can reuse

1Draw the situation.
2Mark the known distance.
3Mark the angle.
4Choose the ratio that connects the known and unknown sides.
5Solve.
6Write the answer with units.

Sketch it in your answer

Draw one observer, one object, one right triangle, and label the given angle at the observer or object point as needed.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

In a one-observer height problem, what is the first thing you should identify after drawing the sketch?

Think first, then show answer

You should identify the right triangle sides: the known horizontal distance, the unknown height, and the given angle. After that, choose the correct trigonometric ratio, usually tan.

Common wrong answer

A common wrong answer is to use the slant side as the height. The slant side is the line of sight, not the vertical height.

Don't mix this up

Students sometimes forget to add the observer's eye height when the question includes it.

Exam tip

Read the question carefully for eye level, ground level, and actual building height. Small wording details change the final answer.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Find the height of a tree, tower, or pole from one observer, or find the distance of the observer from the object.

1
Enough for 1 mark
A one-observer height problem is a trigonometry question where one observer, one angle, and one horizontal distance are used to find a height or distance.
2
Add for 2 marks
One person stands at a known distance from a tower, tree, or pole and measures the angle of elevation or depression.
3
Add for 3 marks
Then one right triangle is formed.

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