medium difficulty5 min

Altitude to hypotenuse relations

In a right triangle, the altitude from the right angle to the hypotenuse creates two smaller triangles that are similar to the original triangle and to each other.

In one line: Altitude is the geometric mean of the split hypotenuse parts.

Practice This Concept

Understand it

Because those triangles are similar, the altitude and the split hypotenuse segments follow special square-type relations. This is one of the most useful results in right-triangle geometry.

Formula and rule

If the hypotenuse is split into p and q, then h² = pq; each leg also satisfies leg² = hypotenuse × adjacent segment.

See it worked out

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

Worked example

If the hypotenuse is split into 16 cm and 9 cm, then h² = 16 × 9 = 144, so h = 12 cm.

Method you can reuse

1Draw the right triangle and altitude.
2Mark the two hypotenuse parts.
3Use h² = pq or the leg relation.
4Solve the value carefully.

Sketch it in your answer

Draw the right triangle, drop the altitude from the right angle to the hypotenuse, and mark the two segments p and q.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What relation connects the altitude and the two parts of the hypotenuse in a right triangle?

Think first, then show answer

The altitude squared equals the product of the two hypotenuse parts, so h² = pq. This relation comes from the similar triangles formed by the altitude.

Common wrong answer

Adding the segments 16 + 9 and calling that the altitude. The relation is multiplicative, not additive.

Don't mix this up

Students often add the two hypotenuse segments or use the wrong side relation.

Exam tip

Label the two parts of the hypotenuse before using h² = pq.

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

The question usually gives hypotenuse segments or one leg and asks for the altitude or another side.

1
Enough for 1 mark
In a right triangle, the altitude from the right angle to the hypotenuse creates two smaller triangles that are similar to the original triangle and to each other.
2
Add for 2 marks
Example: If the hypotenuse is split into 4 cm and 9 cm, then the altitude is 6 cm because h² = 4 × 9.
3
Add for 3 marks
Because those triangles are similar, the altitude and the split hypotenuse segments follow special square-type relations.

MCQ Quiz

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