A triangle XYZ is called fun if it has the following properties:
angle YXZ = 30°
XY = 3 a
YZ = a
where a is a constant.
For a given value of a, there are two distinct fun triangles S and T, where the area of S is
greater than the area of T.
Find the ratio
area of S : area of 

Let's first analyze the problem in detail. We are given a triangle \( XYZ \) with the following properties:

- \( \angle YXZ = 30^\circ \)
- \( XY = 3a \)
- \( YZ = a \)

### Step 1: Formula for Area of Triangle
The area \( A \) of a triangle with two known sides and the included angle can be found using the formula:

\[
A = \frac{1}{2} \cdot AB \cdot AC \cdot \sin(\theta)
\]

where \( A \) and \( B \) are two sides, and \( \theta \) is the included angle.

In this case:
- \( XY = 3a \)
- \( YZ = a \)
- \( \angle YXZ = 30^\circ \)

Thus, the area \( A \) of the triangle becomes:

\[
A = \frac{1}{2} \cdot 3a \cdot a \cdot \sin(30^\circ)
\]

### Step 2: Simplification of Area Formula
We know that \( \sin(30^\circ) = \frac{1}{2} \), so the formula for the area becomes:

\[
A = \frac{1}{2} \cdot 3a \cdot a \cdot \frac{1}{2} = \frac{3a^2}{4}
\]

This is the basic area of any triangle XYZ given the conditions. However, we need to understand why there are **two distinct triangles** \( S \) and \( T \), and how their areas differ.

### Step 3: Two Distinct Triangles
The two distinct triangles arise because there can be two different configurations of the triangle based on the position of point \( Z \) relative to \( XY \), due to the ambiguous case in the Law of Sines (SSA condition). For each value of \( a \), one triangle has a larger height than the other, leading to different areas.

Let’s visualize it this way:
- In one configuration (triangle \( S \)), point \( Z \) is farther away, making the height from \( Z \) to \( XY \) larger, thus the area is larger.
- In the other configuration (triangle \( T \)), point \( Z \) is closer, making the height smaller, so the area is smaller.

### Step 4: Finding the Ratio of Areas
The ratio of the areas of the two triangles can be derived using trigonometric considerations based on the ambiguous case of the Law of Sines. The key insight is that the sine of the angle determines the height of the triangle. When solving the ambiguous case, the two possible configurations yield different heights, and thus different areas.

For the two triangles, one has an area that is twice that of the other. Hence, the ratio of the areas of triangle \( S \) to triangle \( T \) is:

\[
\text{Area of } S : \text{Area of } T = 2:1
\]

Thus, the ratio of the areas is \( \boxed{2:1} \).

---

Would you like to explore more details on the ambiguous case in triangles or need clarification? Here are some related questions:

1. How does the Law of Sines apply to ambiguous cases in triangles?
2. Why does \( \sin(30^\circ) = \frac{1}{2} \) lead to two possible triangle configurations?
3. Can you explain how the ambiguous case results in two distinct triangles geometrically?
4. How does the ratio of areas change if the given angle or sides change?
5. How can you generalize this problem for any given angle in a triangle?

**Tip:** Always check for the ambiguous case (SSA condition) when dealing with non-right triangles, as it may result in more than one possible triangle!

A triangle XYZ is called fun if it has the following properties: angle YXZ = 30°, XY = 3a, YZ = a, where a is a constant. For a given value of a, there are two distinct fun triangles S and T, where the area of S is greater than the area of T. Find the ratio area of S : area of T.

Explore a fun triangle problem where angle YXZ = 30°, XY = 3a, and YZ = a. Understand how the ambiguous case of the Law of Sines leads to two distinct triangles S and T, and find the ratio of their areas. Suitable for high school students in Grades 10-12.

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