https://file.aiquickdraw.com/mathbot/file/1726718590255gq6i14jw.jpg

To solve for \(x\) in the given diagram, we can approach this problem by applying trigonometric identities and the properties of triangles. Here's how we can analyze the problem step by step:

### Step 1: Break Down the Triangles
There are two right triangles in the diagram:
1. The left triangle with angles \(60^\circ\), \(30^\circ\), and a right angle.
2. The upper triangle with angles \(45^\circ\), \(45^\circ\), and a right angle.

### Step 2: Analyze the Upper Triangle
The upper triangle is a 45-45-90 triangle, meaning it is an isosceles right triangle. In such a triangle, the legs are equal, and the hypotenuse is \( \sqrt{2} \) times the length of either leg.

- One leg of this triangle is along the height, and the hypotenuse is given as 15 units.

Thus, the length of the legs can be calculated as:
\[
\text{Leg} = \frac{15}{\sqrt{2}} = \frac{15\sqrt{2}}{2} \approx 10.61 \text{ units}.
\]
This length represents both the height of the full figure and one leg of the 45-45-90 triangle.

### Step 3: Apply to the Left Triangle
The left triangle is a 30-60-90 triangle. In such a triangle, the ratio of the sides opposite the \(30^\circ\), \(60^\circ\), and \(90^\circ\) angles is:
- \(1\) (opposite \(30^\circ\)),
- \(\sqrt{3}\) (opposite \(60^\circ\)),
- \(2\) (hypotenuse).

In this triangle, the height of the triangle is the same as the leg from the previous step: \( \frac{15\sqrt{2}}{2} \).

The base \(x\) of the triangle, opposite the \(30^\circ\) angle, is related to the height (opposite the \(60^\circ\) angle) by the ratio:
\[
\frac{x}{\text{Height}} = \frac{1}{\sqrt{3}}.
\]
Thus:
\[
x = \frac{1}{\sqrt{3}} \times \frac{15\sqrt{2}}{2} = \frac{15\sqrt{6}}{6} \approx 6.12 \text{ units}.
\]

### Final Answer:
The value of \(x\) is approximately \(6.12\) units.

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Would you like further details or explanations on any part of this solution?

### Related Questions:
1. How are trigonometric ratios used in different types of triangles?
2. Can you explain more about the 30-60-90 triangle properties?
3. What are the applications of the Pythagorean theorem in right triangles?
4. How can you calculate the height in a general triangle using trigonometry?
5. What are the properties of a 45-45-90 triangle, and how do they help in solving such problems?

### Tip:
In problems involving special triangles (like 30-60-90 or 45-45-90), remember their side ratios to save time on calculations.

Find the value of x in a triangle with angles 30°, 45°, and 60°, and a given hypotenuse of 15 units.

This problem involves finding the value of x in a triangle with angles of 30°, 45°, and 60°. Using the properties of special triangles, we apply trigonometric formulas to determine x, with a hypotenuse of 15 units. Suitable for Grades 9-12.

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