In a right triangle ABC with right angle at $C$, if $\mathrm{AD}$ is the altitude from vertex $A$ to the hypotenuse $\mathrm{BC}$, which of the following is true about the relationship between the segments $\mathrm{BD}$, $\mathrm{DC}$, and $\mathrm{AD}$?

• A. $\mathrm{AD}\times \mathrm{BC}=\mathrm{BD}\times \mathrm{DC}$
• B. $\mathrm{AD}\times \mathrm{AD}=\mathrm{BD}\times \mathrm{DC}$
• C. $\mathrm{AD}=\mathrm{BD}+\mathrm{DC}$
• D. $\mathrm{AD}=\mathrm{BD}-\mathrm{DC}$

Answer: (B)

Verified step by step guidance

1. Identify the right triangle ABC with the right angle at C, and note that AD is the altitude from vertex A to the hypotenuse BC. This altitude creates two smaller right triangles ABD and ADC within the original triangle.
2. Recall the geometric mean theorem (or altitude-on-hypotenuse theorem), which states that the altitude to the hypotenuse in a right triangle is the geometric mean of the two segments it divides the hypotenuse into. In symbols, this is expressed as \(AD^2 = BD \times DC\).
3. Understand that the hypotenuse BC is divided into two segments by point D, so \(BC = BD + DC\). This is a key relationship but different from the one involving the altitude.
4. Recognize that the product \(AD \times BC = BD \times DC\) is not generally true, so this option can be eliminated based on the theorem.
5. Conclude that the correct relationship is \(AD^2 = BD \times DC\), meaning the square of the altitude equals the product of the two segments of the hypotenuse it creates.