In right-angled △ABC (right at A), an altitude AD is drawn to the hypotenuse BC. Which identity holds?

Introduction / Context:
Dropping an altitude from the right angle to the hypotenuse splits the original right triangle into two smaller right triangles similar to the original and to each other. Several geometric mean relationships follow, including a formula for the altitude length.

Given Data / Assumptions:

• △ABC is right-angled at A.
• AD ⟂ BC, with D on BC.
• Segments on the hypotenuse: BD and DC.

Concept / Approach:

• Similarity: △ABD ~ △ADC ~ △ABC.
• Altitude–segment (geometric mean) theorem: (AD)^2 = BD * DC.
• Other related identities: AB^2 = BD * BC, AC^2 = DC * BC.

Step-by-Step Solution:

Verification / Alternative check:
Coordinate model: Let B(0,0), C(c,0), and A at (x_A, y_A) with right angle at A. Computing using similar triangles again yields AD^2 = BD*DC.

Why Other Options Are Wrong:

• AD^2 = AB × AC or with BD/AB etc. mix unrelated sides; those are not standard altitude relations.
• None of these: Not applicable, since the geometric mean identity is exact.

Common Pitfalls:

• Confusing altitude relations with leg relations AB^2 = BD*BC and AC^2 = DC*BC.

Final Answer:
AD^2 = BD × CD