C
CraftExam
high importancemedium8 min

Combining horizontal distance and height

This concept means using the horizontal distance and the vertical height together in one right triangle to solve a trigonometric problem.

Practice This Concept

Learn the concept

Student-friendly explanation

Many application problems give both a ground distance and a height component, or ask you to combine them after finding one part. The triangle has a horizontal side, a vertical side, and a slant line of sight. Once these are identified correctly, the ratio tan, sin, or cos can be applied depending on which sides are known and which are required. The skill is to see the full geometry before starting the calculation.

How to write this in exams

  1. 1

    Start with the exact idea

    This concept means using the horizontal distance and the vertical height together in one right triangle to solve a trigonometric problem.

  2. 2

    Then show how to use it

    1. Mark all given distances. 2. Separate eye level from ground level if needed. 3. Write the trigonometric ratio. 4. Solve for the missing part. 5. Add or subtract any extra height. 6. State the final answer with units.

  3. 3

    Add one concrete example

    If the distance from a tree is given and the tree height is partly known from eye level, the remaining height can be added or subtracted before using tan.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to forget the eye-level height and report only the vertical part above the observer's eyes.

Definition

This concept means using the horizontal distance and the vertical height together in one right triangle to solve a trigonometric problem.

Example

If the distance from a tree is given and the tree height is partly known from eye level, the remaining height can be added or subtracted before using tan.

Rule to remember

Use tan theta = opposite / adjacent for the triangle, and then adjust the final height if observer eye level is included.

Memory hook

First find the triangle height, then adjust for eye level.

Examples and method

Worked example

An observer's eye is 1.5 m above the ground, and the top of a pole is seen at 30 degrees from 12 m away. First find the height above eye level using tan 30, then add 1.5 m for total height.

Method to apply

1. Mark all given distances. 2. Separate eye level from ground level if needed. 3. Write the trigonometric ratio. 4. Solve for the missing part. 5. Add or subtract any extra height. 6. State the final answer with units.

Diagram support

Draw the ground line, the eye-level horizontal, the vertical object, and label any extra eye height clearly.

How CBSE asks it

Find total height when eye level is given, or find the remaining vertical part after a known measurement is already included.

Avoid common mistakes

Common confusion

Students may use the total height when only the extra height above eye level is needed, or they may ignore the given horizontal distance.

Common wrong answer

A common wrong answer is to forget the eye-level height and report only the vertical part above the observer's eyes.

Exam tip

Check whether the required height is from the ground or only above the observer's eye level. This is a common exam trap.

Quick check

When a problem gives eye level, ground distance, and angle, what should you check before writing the ratio?

You should check whether the height needed is the full height from the ground or only the part above the observer's eye level. That decides whether you add or subtract the eye-level height.

Answer writing and exam use

1-mark answer

This concept means using the horizontal distance and the vertical height together in one right triangle to solve a trigonometric problem.

2-mark answer

This concept means using the horizontal distance and the vertical height together in one right triangle to solve a trigonometric problem. Use tan theta = opposite / adjacent for the triangle, and then adjust the final height if observer eye level is included. If the distance from a tree is given and the tree height is partly known from eye level, the remaining height can be added or subtracted before using tan.

3-mark answer

Many application problems give both a ground distance and a height component, or ask you to combine them after finding one part. The triangle has a horizontal side, a vertical side, and a slant line of sight. Once these are identified correctly, the ratio tan, sin, or cos can be applied depending on which sides are known and which are required. The skill is to see the full geometry before starting the calculation. Use tan theta = opposite / adjacent for the triangle, and then adjust the final height if observer eye level is included. An observer's eye is 1.5 m above the ground, and the top of a pole is seen at 30 degrees from 12 m away. First find the height above eye level using tan 30, then add 1.5 m for total height. Find total height when eye level is given, or find the remaining vertical part after a known measurement is already included. A common wrong answer is to forget the eye-level height and report only the vertical part above the observer's eyes.
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