If $\mathrm{CD}$ is an altitude to the hypotenuse $\mathrm{AB}$ of right triangle $\mathrm{ABC}$ (with right angle at $C$), which statement is necessarily true?

• A. $\frac{\mathrm{CD}}{\mathrm{AB}}=\frac{\mathrm{AC}}{\mathrm{CB}}$
• B. $\mathrm{CD}$ is equal to $\mathrm{AC}$
• C. $\mathrm{CD}$ is less than $\mathrm{AC}$ and $\mathrm{CB}$
• D. $\mathrm{CD}$ is greater than $\mathrm{AB}$

Answer: (A)

Verified step by step guidance

1. Recognize that in right triangle ABC, with the right angle at C, the segment CD is the altitude drawn from the right angle vertex C to the hypotenuse AB.
2. Recall the geometric property that the altitude to the hypotenuse in a right triangle creates two smaller right triangles (ACD and CBD) that are similar to the original triangle ABC and to each other.
3. Use the similarity of triangles to establish relationships between the sides: specifically, the altitude CD relates to the segments AD and DB of the hypotenuse, and the legs AC and BC.
4. Apply the formula for the altitude to the hypotenuse: the length of the altitude CD satisfies the relation \( CD^2 = AD \times DB \), and also the ratios \( \frac{CD}{AB} = \frac{AC}{CB} \) hold due to similarity.
5. Conclude that the statement \( \frac{CD}{AB} = \frac{AC}{CB} \) is necessarily true, while the other options about inequalities or equalities without this ratio are not guaranteed.