Given a right triangle with one leg measuring $15$cm, how many distinct right triangles (up to congruence) can be constructed with this information alone?

• A. $2$
• B. $0$
• C. $\mathrm{Infinitely}\mathrm{many}$
• D. $1$

Answer: (C)

Verified step by step guidance

1. Understand the problem: We are given a right triangle with one leg measuring 15 cm, and we want to determine how many distinct right triangles (up to congruence) can be formed with this information alone.
2. Recall that a right triangle is determined by its two legs and the right angle between them. Knowing only one leg length does not fix the other leg or the hypotenuse.
3. Since the triangle is right-angled, the Pythagorean theorem applies: \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are legs and \(c\) is the hypotenuse. Here, one leg is fixed at 15 cm, but the other leg can vary freely.
4. Because the other leg can be any positive length, each choice of the second leg length creates a different right triangle (not congruent to others), leading to infinitely many possible triangles.
5. Therefore, with only one leg length given, there are infinitely many distinct right triangles that can be constructed.