In triangle ABC, angle BAC = 90° and AD is drawn perpendicular to BC from A. If AD = 6 cm and BD = 4 cm, what is the length (in cm) of the hypotenuse BC?

Introduction / Context:
This geometry question uses special properties of a right angled triangle when an altitude is drawn from the right angle to the hypotenuse. It is a classic configuration that allows you to relate the altitude and the segments of the hypotenuse using simple product relations, without needing trigonometry.

Given Data / Assumptions:

• Triangle ABC has angle BAC = 90°, so A is the right angle.
• BC is the hypotenuse.
• AD is perpendicular to BC, so AD is the altitude from the right angle to the hypotenuse.
• AD = 6 cm.
• BD = 4 cm, where D lies on BC between B and C.
• We must find the full length BC.

Concept / Approach:
In a right angled triangle, when you drop an altitude from the right angle to the hypotenuse, several useful relations hold: AD^2 = BD * CD BD + CD = BC These allow you to find the unknown segment CD from the product relation, and then BC by adding BD and CD. This approach is often faster than coordinate methods or repeated use of Pythagoras theorem on multiple smaller triangles.

Step-by-Step Solution:
Step 1: Use the relation AD^2 = BD * CD for a right triangle with altitude to the hypotenuse. Step 2: Substitute AD = 6 cm and BD = 4 cm. Step 3: So 6^2 = 4 * CD, which gives 36 = 4 * CD. Step 4: Solve for CD: CD = 36 / 4 = 9 cm. Step 5: The hypotenuse BC is the sum of BD and CD, so BC = BD + CD = 4 + 9 = 13 cm.

Verification / Alternative check:
We can verify by reconstructing triangle ABC. One leg squared is AB^2 = BD * BC = 4 * 13 = 52. The other leg squared is AC^2 = CD * BC = 9 * 13 = 117. The sum AB^2 + AC^2 = 52 + 117 = 169, which is equal to BC^2 = 13^2 = 169. This confirms that BC = 13 cm is consistent with the Pythagoras theorem for the entire triangle ABC.

Why Other Options Are Wrong:
10 cm, 12 cm, 15 cm and 11 cm do not satisfy the altitude relation AD^2 = BD * CD when BD = 4 cm and AD = 6 cm. If BC were one of these values, the derived CD and the subsequent checks with Pythagoras theorem would fail, so these options are not compatible with the given information.

Common Pitfalls:
Students often try to use Pythagoras theorem directly on smaller triangles without recalling the special product relations, which makes the problem longer. Another mistake is to assume BD and CD are equal, which is not generally true. Remembering the identity AD^2 = BD * CD in right triangles with an altitude to the hypotenuse helps solve such problems quickly and accurately.

Final Answer:
The length of BC is 13 cm.