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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

    If the hypotenuse is split into 4 cm and 9 cm, then the altitude is 6 cm because h² = 4 × 9.

  4. 4

    Avoid this incomplete answer

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

Definition

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.

Example

If the hypotenuse is split into 4 cm and 9 cm, then the altitude is 6 cm because h² = 4 × 9.

Rule to remember

If the hypotenuse is split into p and q, then h^2 = pq; each leg also satisfies leg^2 = hypotenuse × adjacent segment.

Memory hook

Altitude is the geometric mean of the split hypotenuse parts.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

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.

Answer writing and exam use

1-mark answer

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-mark answer

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. If the hypotenuse is split into p and q, then h^2 = pq; each leg also satisfies leg^2 = hypotenuse × adjacent segment. If the hypotenuse is split into 4 cm and 9 cm, then the altitude is 6 cm because h² = 4 × 9.

3-mark answer

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. If the hypotenuse is split into p and q, then h^2 = pq; each leg also satisfies leg^2 = hypotenuse × adjacent segment. If the hypotenuse is split into 16 cm and 9 cm, then h² = 16 × 9 = 144, so h = 12 cm. The question usually gives hypotenuse segments or one leg and asks for the altitude or another side. Adding the segments 16 + 9 and calling that the altitude. The relation is multiplicative, not additive.
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