medium difficulty7 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.

In one line: First find the triangle height, then adjust for eye level.

Practice This Concept

Understand it

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.

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.

Formula and rule

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

See it worked out

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

Worked example

1An 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.
2First find the height above eye level using tan 30, then add 1.5 m for total height.

Method you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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.

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.

Don't mix this up

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

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.

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

1
Enough for 1 mark
This concept means using the horizontal distance and the vertical height together in one right triangle to solve a trigonometric problem.
2
Add for 2 marks
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.
3
Add for 3 marks
The triangle has a horizontal side, a vertical side, and a slant line of sight.

MCQ Quiz

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