In right triangle ABC, right angled at B, BD is an altitude to hypotenuse AC. If AD = 9 cm and DC = 16 cm, what is the length of BD (in cm)?

Introduction / Context:
This problem uses a special property of right triangles when an altitude is drawn from the right angle to the hypotenuse. That altitude divides the hypotenuse into two segments and creates several useful relationships between the segments, the altitude, and the sides of the triangle. Recognising and applying these relationships can simplify what might otherwise be a lengthy Pythagorean calculation.

Given Data / Assumptions:

• Triangle ABC is a right triangle with right angle at B.
• BD is the altitude drawn from B to the hypotenuse AC.
• The hypotenuse AC is divided into segments AD = 9 cm and DC = 16 cm.
• We are asked to find the length of altitude BD.

Concept / Approach:
In a right triangle, when an altitude is drawn from the right angle to the hypotenuse, there is a key relationship:
BD^2 = AD * DC This comes from the similarity of the smaller triangles formed with the original triangle. Using this relationship is often faster than working with the side lengths of all three triangles separately.

Step-by-Step Solution:
Step 1: Identify the given segments on the hypotenuse: AD = 9 cm and DC = 16 cm. Step 2: Use the altitude relationship BD^2 = AD * DC. Step 3: Substitute the given values: BD^2 = 9 * 16. Step 4: Compute the product: 9 * 16 = 144. Step 5: Take the square root of both sides to find BD: BD = sqrt(144) = 12. Step 6: Therefore, the altitude BD is 12 cm long.

Verification / Alternative check:
An alternative check uses the fact that AB^2 = AD * AC and CB^2 = DC * AC, where AC = AD + DC = 9 + 16 = 25 cm. Using these, we obtain AB^2 = 9 * 25 = 225, so AB = 15 cm, and CB^2 = 16 * 25 = 400, so CB = 20 cm. Applying the Pythagorean theorem directly with AB = 15 and CB = 20 gives AC^2 = 15^2 + 20^2 = 225 + 400 = 625, so AC = 25 cm, consistent with 9 + 16. Then the area can be written as (1/2) * AB * CB = (1/2) * 15 * 20 = 150, and also as (1/2) * AC * BD = (1/2) * 25 * BD. Setting these equal gives 150 = (1/2) * 25 * BD, so BD = 12, confirming our answer.

Why Other Options Are Wrong:

• 6 and 9: These are smaller values and correspond to taking the square root of only one of the segments or making arithmetic errors.
• 18 and 21: These are larger than the altitude that fits in this triangle and do not satisfy the relation BD^2 = 9 * 16.

Common Pitfalls:
Some students incorrectly assume BD is the average or sum of AD and DC. Others forget the altitude formula and try complicated algebra. Remember the simple relation BD^2 = AD * DC whenever an altitude from a right angle is drawn to the hypotenuse; it is a powerful shortcut for many right triangle problems.

Final Answer:
Hence, the length of BD is 12 cm.