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.