C
CraftExam
high importancemedium8 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.

Practice This Concept

Learn the concept

Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

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

Definition

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

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.

Rule to remember

If two observation points are on the same straight line, then tan theta1 = h/d1 and tan theta2 = h/d2 provide the pair of equations.

Memory hook

Two observers, two triangles, one shared height.

Examples and method

Worked example

From one point 20 m away, the angle of elevation is 30 degrees. From another point 10 m closer, the angle is 60 degrees. Using two tan equations with the same height gives a solvable pair.

Method to apply

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

Diagram support

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

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.

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

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.

Answer writing and exam use

1-mark answer

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-mark answer

A two-observer distance problem is a trigonometry question where two viewing positions or two angles are used to find a height or distance. If two observation points are on the same straight line, then tan theta1 = h/d1 and tan theta2 = h/d2 provide the pair of equations. 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.

3-mark answer

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. If two observation points are on the same straight line, then tan theta1 = h/d1 and tan theta2 = h/d2 provide the pair of equations. From one point 20 m away, the angle of elevation is 30 degrees. From another point 10 m closer, the angle is 60 degrees. Using two tan equations with the same height gives a solvable pair. Find the height of a tower using two angles from two points, or find the distance between two points using a common height. A common wrong answer is to use only one triangle and ignore the second observation point. That loses half the information.
MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept