In right triangle ABC, the right angle is at B (∠B = 90°). BD is drawn from B perpendicular to the hypotenuse AC, so BD is an altitude. The segments on AC satisfy AD = 9 cm and DC = 16 cm. What is the length (in cm) of the altitude BD?

Introduction:
This problem uses the special relationships in a right triangle between the hypotenuse and the altitude drawn from the right angle to the hypotenuse. These relationships connect the altitude length with the segments into which it divides the hypotenuse.

Given Data / Assumptions:

• Triangle ABC is right angled at B.
• BD is the altitude from B to the hypotenuse AC, so BD ⟂ AC.
• On the hypotenuse, AD = 9 cm and DC = 16 cm.
• We must find BD.

Concept / Approach:
In a right triangle, if an altitude is drawn from the right angle to the hypotenuse, then the square of the altitude equals the product of the segments into which it divides the hypotenuse. That is, BD² = AD * DC. This is a standard result from similar triangles or geometric mean properties and provides a direct way to compute the altitude without finding all side lengths.

Step-by-Step Solution:
We know AD = 9 cm and DC = 16 cm. Using the relation BD² = AD * DC, we get BD² = 9 * 16. Compute 9 * 16 = 144. So BD² = 144. Take the positive square root (since length is positive): BD = √144 = 12 cm.

Verification / Alternative check:
Alternatively, one could find the full hypotenuse AC = AD + DC = 25 cm and then show that triangles formed are similar and that the altitude is the geometric mean of the segments. This again leads to BD² = 9 * 16 = 144 and BD = 12 cm, confirming the result.

Why Other Options Are Wrong:
Values 6, 18, and 21 cm do not satisfy BD² = 144. For example, 6² = 36, 18² = 324, 21² = 441. Only 12² equals 144, so only 12 cm is consistent with the given segment lengths on the hypotenuse.

Common Pitfalls:
A common error is to assume BD is the arithmetic mean of AD and DC or to apply the Pythagorean theorem incorrectly to non right angles. Remember that the geometric mean relationship BD² = AD * DC specifically holds for the altitude drawn from the right angle to the hypotenuse.

Final Answer:
The length of BD is 12 cm.