The problem involves solving a right triangle and calculating its area.

Step 1: Understand the Given Triangle

The triangle is a right triangle with:

• One angle of $90^\circ$,
• Another angle of $30^\circ$,
• The hypotenuse (the side opposite the $90^\circ$ angle) is $25$ meters,
• One leg adjacent to the $30^\circ$ angle is $24$ meters.

Step 2: Find the Missing Side Length

Let's denote the sides as follows:

• $a$ is the side opposite the $30^\circ$ angle,
• $b$ is the side opposite the $60^\circ$ angle,
• $c = 25$ meters is the hypotenuse.

Using the Pythagorean theorem: $c^2 = a^2 + b^2$

Given:

• $c = 25$ meters,
• $b = 24$ meters,

Substitute the known values: $25^2 = a^2 + 24^2$ $625 = a^2 + 576$ $a^2 = 625 - 576 = 49$ $a = \sqrt{49} = 7 \text{ meters}$

So, the missing side $a$ is $7$ meters.

Step 3: Calculate the Area of the Triangle

The area $A$ of a right triangle is given by: $A = \frac{1}{2} \times \text{base} \times \text{height}$ Here, base = 24 meters, and height = 7 meters. $A = \frac{1}{2} \times 24 \times 7 = \frac{1}{2} \times 168 = 84 \text{ square meters}$

Final Answer:

The area of the triangle is $84$ square meters.

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Would you like more details or have any other questions?

Here are some related questions to consider:

1. How do you determine the other angle in the triangle?
2. Can the area formula be applied to any type of triangle?
3. What is the significance of a $30^\circ$-$60^\circ$-$90^\circ$ triangle?
4. How do you calculate the area if the triangle is not a right triangle?
5. What would change if the sides were given in different units?
6. Can trigonometric ratios be used to solve the triangle?
7. How does the Pythagorean theorem apply to different types of triangles?
8. What would happen if the hypotenuse length was unknown?

Tip: For right triangles, remembering special angle relationships like $30^\circ$-$60^\circ$-$90^\circ$ can simplify calculations significantly.