Triangle ABC is right angled at B. BD is the altitude drawn from B to hypotenuse AC. If DC = 9 cm and AC = 25 cm, what is the length of side BC (in cm)?

Introduction / Context:
This question uses a special property of right angled triangles and the altitude from the right angle to the hypotenuse. In triangle ABC, angle B is 90 degrees and BD is the altitude from B to the hypotenuse AC. You are given the length of the hypotenuse AC and the segment DC on it, and asked to find the length of side BC. This tests your knowledge of geometric mean relationships in right triangles and the Pythagoras theorem.

Given Data / Assumptions:

- Triangle ABC is right angled at B. - BD is drawn from B perpendicular to AC, so BD is an altitude. - AC is the hypotenuse and has length 25 cm. - DC, a segment of the hypotenuse, has length 9 cm. - We need to find the length of side BC.

Concept / Approach:
In a right triangle, when an altitude is drawn from the right angle to the hypotenuse, several useful relationships hold. One of them states that the square of each leg equals the product of the hypotenuse and the projection of that leg onto the hypotenuse. Specifically, BC^2 = AC * DC and AB^2 = AC * AD, where AD and DC are the segments into which the altitude divides the hypotenuse. We use this geometrical mean relation directly since AC and DC are given.

Step-by-Step Solution:
Step 1: Note that AC is the hypotenuse of the right triangle ABC and AC = 25 cm. Step 2: BD is the altitude from the right angle B to hypotenuse AC, dividing AC into two segments AD and DC. Step 3: We are given DC = 9 cm. Then the other segment AD is AC - DC = 25 - 9 = 16 cm (this may be useful for cross checking). Step 4: Use the right triangle property: The square of the leg adjacent to DC, which is BC, satisfies BC^2 = AC * DC. Step 5: Substitute the known values: BC^2 = 25 * 9. Step 6: Compute 25 * 9 = 225. Step 7: So BC^2 = 225. Taking the positive square root, BC = √225 = 15 cm.

Verification / Alternative check:
We can also verify by finding AB using the related formula AB^2 = AC * AD = 25 * 16 = 400, so AB = 20 cm. Check Pythagoras theorem in triangle ABC: AB^2 + BC^2 = 20^2 + 15^2 = 400 + 225 = 625. The square root of 625 is 25, equal to AC. This confirms that the side lengths 15, 20 and 25 form a right triangle and that BC = 15 cm is correct.

Why Other Options Are Wrong:
Values like 12, 18 or 16 do not satisfy the relationship BC^2 = 25 * 9. For example, if BC were 12, BC^2 would be 144, which is not equal to 225. Similarly, 16^2 = 256 and 18^2 = 324 both fail to match 225. Thus these options contradict the geometric mean property and Pythagoras theorem for the given triangle.

Common Pitfalls:
Students sometimes attempt to use Pythagoras theorem directly without recognizing the role of the altitude and the segments on the hypotenuse, which leads to missing information. Others may incorrectly apply the formula BC^2 = AD * DC instead of AC * DC. Remember that the leg squared equals hypotenuse times the adjacent hypotenuse segment, while the altitude squared equals the product of the two hypotenuse segments. Keeping these relationships straight is essential.

Final Answer:
The length of side BC is 15 cm, which corresponds to option D.