To find the exact area, let's solve for \( a \) using the given equation:
\[ \sqrt{36 + a^2} + a\sqrt{10} = 25 - 5a \]

To solve this equation, we will square both sides to eliminate the square root. However, this is a non-linear equation, so it will require some algebraic manipulation:

1. Starting with:
   \[ \sqrt{36 + a^2} + a\sqrt{10} = 25 - 5a \]

2. Isolate the square root:
   \[ \sqrt{36 + a^2} = 25 - 5a - a\sqrt{10} \]

3. Square both sides to eliminate the square root:
   \[ 36 + a^2 = (25 - 5a - a\sqrt{10})^2 \]

Expand the right side:
\[
(25 - 5a - a\sqrt{10})^2 = 625 - 250a - 50a\sqrt{10} + 25a^2 + 10a^2 + a^2(\sqrt{10})^2
\]
\[
= 625 - 250a - 50a\sqrt{10} + 25a^2 + 10a^2 + 10a^2
\]
\[
= 625 - 250a - 50a\sqrt{10} + 45a^2
\]

4. Set the expanded equation equal to the left side:
\[
36 + a^2 = 625 - 250a - 50a\sqrt{10} + 45a^2
\]

5. Combine like terms:
\[
36 + a^2 = 625 - 250a - 50a\sqrt{10} + 45a^2
\]

6. Move all terms to one side to set the equation to zero:
\[
0 = 589 - 250a - 50a\sqrt{10} + 44a^2
\]

Since solving this analytically is complex, we'll use numerical methods to solve for \(a\).

Let's use a numerical solver to find the value of \(a\):

\[
589 - 250a - 50a\sqrt{10} + 44a^2 = 0
\]

To solve this equation numerically, I will use Python. Let's proceed with this step.The solutions to the equation are:
\[ a = \frac{25\sqrt{10} + 125 + \sqrt{-4041 + 6250\sqrt{10}}}{44} \]
\[ a = \frac{25\sqrt{10} + 125 - \sqrt{-4041 + 6250\sqrt{10}}}{44} \]

Since \(a\) must be a positive real number and the expression under the square root \(\sqrt{-4041 + 6250\sqrt{10}}\) must be non-negative, we should consider only the real, positive solution.

To simplify, let's approximate the numerical value of \( a \).

I'll proceed with finding the numerical value of the valid solution.The numerical solution for \( a \) is approximately \( 7.487 \).

Now, let's calculate the areas of the shaded triangles using this value:

1. **Area of the first triangle**:
   \[
   \text{Area} = \frac{1}{2} \times 6 \times 7.487 = 3 \times 7.487 = 22.461
   \]

2. **Area of the second triangle**:
   \[
   \text{Area} = \frac{1}{2} \times 3 \times (7.487)^2 = \frac{3 \times 56.04}{2} = 84.06
   \]

So, the total area of the shaded triangles is:
\[
\text{Total shaded area} = 22.461 + 84.06 = 106.521
\]

The exact area of the two shaded right-angled triangles is approximately \( 106.521 \) square units.

Would you like any further clarifications or details? Here are some related questions you might be interested in:

1. How can we derive the equation for the perimeter in such problems?
2. What are the typical methods to solve transcendental equations?
3. How do numerical solvers work to find solutions for complex equations?
4. Can we use other forms of numerical analysis to verify the value of \( a \)?
5. How do we ensure the accuracy of numerical approximations in mathematical problems?
6. Are there alternative geometric methods to solve this type of problem?
7. What are some common challenges when solving equations with square roots?
8. How does the value of \( a \) affect the overall geometry of the problem?

**Tip:** Always check the reasonableness of your solution by verifying it with the problem's constraints. This ensures that approximations and numerical solutions are within expected bounds.

Learn how to find the exact area of two shaded right-angled triangles by solving a complex equation step-by-step. This problem involves equations with square roots, numerical methods for solving non-linear equations, and geometric applications. Suitable for advanced high school students and above.

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