Key Concepts and Formulas

• Area of a Triangle with Vertices at Origin and on Coordinate Axes: The area of a triangle with vertices at $O(0,0)$, $A(x_1, 0)$, and $B(0, y_1)$ is given by $\text{Area} = \frac{1}{2} |x_1 \cdot y_1|$.
• Number of Divisors of an Integer: If $N = p_1^{a_1} p_2^{a_2} \cdots p_r^{a_r}$ is the prime factorization of $N$, then the number of positive divisors is $\tau(N) = (a_1+1)(a_2+1)\cdots(a_r+1)$.

Step-by-Step Solution

Step 1: Define Vertices and Set Up the Area Equation

Let the vertices of a triangle be $O(0,0)$, $A(x_1, 0)$, and $B(0, y_1)$, where $x_1, y_1 \in \mathbb{Z}$ and $x_1, y_1 \neq 0$. The area is given as 50. Using the area formula: $\frac{1}{2} |x_1 \cdot y_1| = 50$ Multiplying both sides by 2: $|x_1 \cdot y_1| = 100$ Why this step? We translate the problem's geometric condition (area) into an algebraic equation. The absolute value is essential since coordinates can be negative, but area is always non-negative.

Step 2: Analyze the Product $x_1 \cdot y_1$

The equation $|x_1 \cdot y_1| = 100$ implies two cases:

1. $x_1 \cdot y_1 = 100$
2. $x_1 \cdot y_1 = -100$

Why this step? The absolute value leads to two possibilities, each leading to different $(x_1, y_1)$ pairs representing distinct triangles.

Step 3: Count Integer Pairs for $x_1 \cdot y_1 = 100$

For $x_1 \cdot y_1 = 100$, both $x_1$ and $y_1$ must have the same sign. The prime factorization of 100 is $100 = 2^2 \cdot 5^2$. The number of positive divisors of 100 is $\tau(100) = (2+1)(2+1) = 9$.

• Subcase 3a: Both $x_1$ and $y_1$ are positive integers. There are 9 positive divisors, so there are 9 pairs $(x_1, y_1)$ with $x_1 > 0$ and $y_1 > 0$.

• Subcase 3b: Both $x_1$ and $y_1$ are negative integers. There are 9 negative divisors, so there are 9 pairs $(x_1, y_1)$ with $x_1 < 0$ and $y_1 < 0$.

The total number of pairs for $x_1 \cdot y_1 = 100$ is $9 + 9 = 18$. Why this step? We systematically enumerate integer possibilities. Since the product is positive, both integers share the same sign. The number of positive divisors gives us the count for positive pairs, and consequently, for negative pairs.

Step 4: Count Integer Pairs for $x_1 \cdot y_1 = -100$

For $x_1 \cdot y_1 = -100$, $x_1$ and $y_1$ must have opposite signs.

• Subcase 4a: $x_1$ is positive and $y_1$ is negative. There are 9 positive divisors for $x_1$, each paired with a corresponding negative $y_1$. This gives 9 pairs.

• Subcase 4b: $x_1$ is negative and $y_1$ is positive. There are 9 negative divisors for $x_1$, each paired with a corresponding positive $y_1$. This gives 9 pairs.

The total number of pairs for $x_1 \cdot y_1 = -100$ is $9 + 9 = 18$. Why this step? Similar to Step 3, we cover all integer possibilities. When the product is negative, the integers have opposite signs.

Step 5: Calculate the Total Number of Elements in Set S

The total number of distinct triangles is the sum of the pairs: $\text{Total Triangles} = (\text{Pairs for } x_1 y_1 = 100) + (\text{Pairs for } x_1 y_1 = -100)$ $\text{Total Triangles} = 18 + 18 = 36$

Common Mistakes & Tips

• Missing the Absolute Value: Forgetting the absolute value in the area formula leads to missing half the possible triangles.
• Assuming Positive Integers Only: Remember that "integral coordinates" means both positive and negative integers.
• Not counting distinct triangles: Each unique $(x_1, y_1)$ defines a different triangle.

Summary

We found the number of triangles with area 50 and vertices at the origin and on the coordinate axes with integer coordinates by considering both positive and negative values for $x_1$ and $y_1$. The absolute value in the area formula gives two cases: $x_1y_1 = 100$ and $x_1y_1 = -100$. The number of integer pairs for each case is determined by the number of divisors of 100, leading to a total of 36 distinct triangles.

The final answer is $\boxed{36}$, which corresponds to option (C).