Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the othe

Q: Wait, why do we use absolute values in the area formula?

Great question! The area of a triangle is a physical quantity, which must always be positive. Since the coordinates $a$ and $b$ can be negative (placing the triangle in different quadrants), we use $|a|$ and $|b|$ to represent the lengths of the base and height, ensuring the area calculation remains physically meaningful.

Q: Why isn't the answer just 9? I calculated the divisors of 100 and got 9.

You are absolutely right that there are 9 pairs of positive integers $(|a|, |b|)$ that satisfy the equation. However, the problem asks for the number of triangles in the $xy$-plane. Each pair of magnitudes can be placed in any of the four quadrants by changing the signs of $a$ and $b$. Thus, for each magnitude pair, there are 4 distinct triangles!

Q: What if $a$ or $b$ is zero? Does that change the number of triangles?

The problem states that the vertices lie on the coordinate axes and form a triangle. If $a$ or $b$ were zero, the vertices would coincide with the origin, collapsing the triangle into a line segment. Therefore, $a$ and $b$ must be non-zero integers, which is why we exclude zero from our divisor count.

Question

Let $S$ be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area 50 sq. units, then the number of elements in the set $S$ is:

Options:

(A) 9

(B) 18

(C) 32

(D) 36

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with finding the number of triangles in the set $S$, where each triangle has one vertex at the origin $(0,0)$ and the other two vertices on the coordinate axes with integral coordinates. The area of each triangle is fixed at $50$ square units.

Let the vertices of the triangle be $O(0,0)$, $A(a,0)$, and $B(0,b)$, where $a, b \in \mathbb{Z} \setminus \{0\}$. This configuration forms a right-angled triangle with the right angle at the origin.

The base of this triangle is the segment $OA$ with length $|a|$, and the height is the segment $OB$ with length $|b|$. The area of a right-angled triangle is given by:

$$	ext{Area} = \frac{1}{2} 	imes |a| 	imes |b|$$

### The Master Equation

Given that the area is $50$, we substitute this into our formula:

$$\frac{1}{2} |a| |b| = 50$$

Multiplying both sides by $2$, we arrive at the fundamental relation:

$$|a| |b| = 100$$

We are now looking for the number of integer pairs $(a, b)$ such that the product of their absolute values is $100$. This transforms our geometric problem into a classic number theory challenge.

### The Divisor Hunt

To find the number of pairs $(|a|, |b|)$, we must count the divisors of $100$. First, we perform the prime factorization of $100$:

$$100 = 10^2 = (2 	imes 5)^2 = 2^2 	imes 5^2$$

The formula for the total number of divisors of a number $N = p_1^{x} p_2^{y}$ is $(x+1)(y+1)$. Here, our exponents are $2$ and $2$.

Thus, the number of positive divisors is:

$$(2+1)(2+1) = 3 	imes 3 = 9$$

This means there are exactly $9$ pairs of positive integers $(|a|, |b|)$ that satisfy the area equation.

### Accounting for Quadrants

The coordinate plane consists of four quadrants. For every pair of magnitudes $(|a|, |b|)$, we can choose the signs of $a$ and $b$ independently.

Specifically, for each pair, we can have:
1. $(+|a|, +|b|)$ in the first quadrant.
2. $(-|a|, +|b|)$ in the second quadrant.
3. $(-|a|, -|b|)$ in the third quadrant.
4. $(+|a|, -|b|)$ in the fourth quadrant.

Each choice creates a unique triangle in the $xy$-plane. Therefore, for each of our $9$ magnitude pairs, there are $4$ possible triangles.

### Final Calculation

The total number of elements in set $S$ is calculated as:

$$9 	imes 4 = 36$$

By carefully translating the geometric constraints into an algebraic equation and accounting for the symmetry of the coordinate plane, we conclude that the total number of such triangles is **36**.

Select Answer:

Visualized Solution

Defining the Triangle Vertices

Area of the Right-Angled Triangle

Setting Up the Area Equation

Simplifying the Equation

Prime Factorization of $100$

Calculating Total Divisors

Accounting for the Four Quadrants

Final Calculation of Set $S$

The Sigma Insight: Area of Triangle

Analyzing the Setup

The Master Equation

The Divisor Hunt

Accounting for Quadrants

Final Calculation

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