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To find the exact value of \( x \) in the given right triangle, let's analyze the information provided in the diagram:

### Given Information:
- The base of the right triangle is 100 m.
- The height is \( x \) m.
- There are two angles inside the triangle: \( 45^\circ \) and \( 30^\circ \).

### Objective:
We need to find the exact value of \( x \).

### Approach:

#### Step 1: Analyzing the triangle
There are two smaller triangles within the larger right triangle:
1. A 45-45-90 triangle.
2. A 30-60-90 triangle.

Since \( x \) is the height of the large triangle and the common side for both smaller triangles, let's use trigonometric identities.

### Step 2: Use the 45-45-90 Triangle to find part of the base

In the 45-45-90 triangle, the legs are equal. Thus, if we denote the length of the segment adjacent to \( 45^\circ \) as \( y \) (which is part of the 100 m base), we have:
\[
\text{tan}(45^\circ) = \frac{\text{opposite (height)}}{\text{adjacent}} = \frac{x}{y}
\]
Since \( \text{tan}(45^\circ) = 1 \):
\[
x = y
\]

This means that the segment of the base adjacent to the \( 45^\circ \) angle is also \( x \) m.

#### Step 3: Apply the 30-60-90 Triangle properties

In the 30-60-90 triangle:
- The side opposite the \( 30^\circ \) angle is half the hypotenuse.
- The side opposite the \( 60^\circ \) angle (which is \( x \)) is \( \sqrt{3}/2 \) times the hypotenuse.

Let the segment of the base adjacent to the \( 30^\circ \) angle be \( z \), and since the total base is 100 m:
\[
z + y = 100
\]
But since \( y = x \), we have:
\[
z + x = 100
\]
Now, using the 30-60-90 triangle:
\[
\text{tan}(30^\circ) = \frac{x}{z}
\]
We know that \( \text{tan}(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[
\frac{1}{\sqrt{3}} = \frac{x}{z}
\]
\[
z = x\sqrt{3}
\]
Substitute this into the equation \( z + x = 100 \):
\[
x\sqrt{3} + x = 100
\]
Factor out \( x \):
\[
x(\sqrt{3} + 1) = 100
\]
Solve for \( x \):
\[
x = \frac{100}{\sqrt{3} + 1}
\]
To rationalize the denominator:
\[
x = \frac{100(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{100(\sqrt{3} - 1)}{3 - 1} = \frac{100(\sqrt{3} - 1)}{2}
\]
Simplifying:
\[
x = 50(\sqrt{3} - 1) \text{ meters}
\]

### Final Answer:
\[
x = 50(\sqrt{3} - 1) \text{ meters}
\]

Would you like any further details or have any questions?

### Relative Questions:
1. How can you verify the correctness of the calculated height \( x \) using different trigonometric methods?
2. What would happen to the value of \( x \) if the base was increased or decreased?
3. How can the properties of the 45-45-90 and 30-60-90 triangles be applied in other geometry problems?
4. If the height \( x \) were given, how could you find the length of the hypotenuse of the large triangle?
5. What is the significance of rationalizing the denominator in the final expression for \( x \)?

### Tip:
When dealing with triangles involving special angles (like \(30^\circ\), \(45^\circ\), and \(60^\circ\)), always consider using their specific trigonometric ratios to simplify the problem-solving process.

Learn how to find the exact value of x in a right triangle problem involving special triangles and trigonometric ratios. Detailed step-by-step solution provided. Suitable for high school students in Grades 10-12.

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