medium difficulty8 min

Two-observer distance problem

A two-observer distance problem is a trigonometry question where two viewing positions or two angles are used to find a height or distance.

In one line: Two observers, two triangles, one shared height.

Practice This Concept

Understand it

In these problems, the object is viewed from two different points or by two observers.
The key idea is to make two right triangles that share the same height or object position.
Since the height is common, we write two equations and solve them together.
These questions often use angles of elevation from two points on the same straight line, such as two positions on the ground or two buildings.

Example

If two students stand at different distances from a tower and see the top at different angles, the tower height can be found by forming two tan equations and solving them.

Formula and rule

If two observation points are on the same straight line, then tan θ1 = h/d1 and tan θ2 = h/d2 provide the pair of equations.

See it worked out

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

Worked example

From one point 20 m away, the angle of elevation is 30 degrees.

1From another point 10 m closer, the angle is 60 degrees.
2Using two tan equations with the same height gives a solvable pair.

Method you can reuse

1Draw both observation points.
2Mark the common height.
3Write tan for each triangle.
4Form two equations.
5Solve them together.
6Check whether the final answer is reasonable.

Sketch it in your answer

Draw two observation points on a line with the same object, and make two right triangles sharing the same vertical height.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why are two equations usually needed in a two-observer height problem?

Think first, then show answer

Two equations are needed because the same height is connected to two different distances or two different angles. We use the common height to link both triangles and solve the problem.

Common wrong answer

A common wrong answer is to use only one triangle and ignore the second observation point. That loses half the information.

Don't mix this up

Students may forget that the height is the same in both triangles and write two unrelated equations.

Exam tip

Look for the common height or common horizontal distance first. That shared value is usually the bridge between the two equations.

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 tower using two angles from two points, or find the distance between two points using a common height.

1
Enough for 1 mark
A two-observer distance problem is a trigonometry question where two viewing positions or two angles are used to find a height or distance.
2
Add for 2 marks
In these problems, the object is viewed from two different points or by two observers.
3
Add for 3 marks
The key idea is to make two right triangles that share the same height or object position.

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