In right triangle ABC, angle B is 90° and BD is the altitude drawn from B to hypotenuse AC. If AD = 4 cm and DC = 12 cm, what is the length (in centimetres) of side AB?

Introduction / Context:
This problem uses a special set of relationships in a right triangle when an altitude is drawn from the right angle to the hypotenuse. The given data are the segments into which the hypotenuse is divided by the foot of the altitude. Using known geometric properties and formulae, we can compute the length of a leg of the triangle. This is a classic question in Euclidean geometry and appears frequently in aptitude and competitive exams.

Given Data / Assumptions:
- Triangle ABC is right angled at B.- BD is the altitude from B to the hypotenuse AC.- AD = 4 cm and DC = 12 cm, where D lies on AC between A and C.- We need to find the length of AB.- Standard properties of right triangles and altitude to the hypotenuse apply.

Concept / Approach:
In a right triangle with altitude drawn from the right angle to the hypotenuse, there is a relationship between the legs, the hypotenuse, and the segments into which the hypotenuse is divided. If AC is the hypotenuse and D is the foot of the altitude, then AC = AD + DC. One of the key formulae is AB^2 = AD * AC, where AB is the leg adjacent to segment AD. Once we compute the full hypotenuse AC from AD and DC, we can apply this relationship to find AB. This reduces the problem to simple multiplication and taking a square root.

Step-by-Step Solution:
Step 1: Given AD = 4 cm and DC = 12 cm.Step 2: Compute the hypotenuse AC = AD + DC = 4 + 12 = 16 cm.Step 3: Use the right triangle altitude property: AB^2 = AD * AC.Step 4: Substitute AD = 4 and AC = 16.Step 5: AB^2 = 4 * 16 = 64.Step 6: Take the square root: AB = √64 = 8 cm.Step 7: Therefore, the length of side AB is 8 centimetres.

Verification / Alternative check:
We can also find BC using the similar relation BC^2 = DC * AC = 12 * 16 = 192, giving BC = √192 = √(64 * 3) = 8√3. Then check that AC^2 = AB^2 + BC^2. AC^2 should be 16^2 = 256. Compute AB^2 + BC^2 = 64 + (8√3)^2 = 64 + 64 * 3 = 64 + 192 = 256, which matches AC^2. This confirms that AB = 8 cm is consistent with the Pythagorean theorem and altitude properties.

Why Other Options Are Wrong:
- 9: Squaring 9 gives 81, which would require AD * AC = 81, contradicting the given segments.- 10: This would give AB^2 = 100, not equal to AD * AC = 64.- 6: Squaring 6 gives 36, again not matching AD * AC = 64.- 12: Squaring 12 gives 144, which is also inconsistent with the product 4 * 16.

Common Pitfalls:
Some students mistakenly apply the Pythagorean theorem directly using AD and DC as legs, which is incorrect because these are segments of the hypotenuse, not the legs of the triangle. Others might confuse the specific altitude relations and attempt to use AB^2 = AD^2 + AC^2, which is not valid. Remembering the exact property AB^2 = AD * AC is crucial in solving such altitude related questions efficiently.

Final Answer:
The length of side AB is 8 centimetres.