From the provided image, it appears that the task involves solving for $a$, the side length of an equilateral triangle, given its height $h = 10 \, \text{cm}$. The problem uses the Pythagorean theorem as part of the solution process.

Step-by-step Explanation:

1. Triangle Breakdown:
   In an equilateral triangle, the height $h$ divides the triangle into two right triangles. Each right triangle has:

   • One leg = $\frac{a}{2}$ (half the base of the triangle),
   • The other leg = $h = 10 \, \text{cm}$ (the given height),
   • The hypotenuse = $a$ (the side of the equilateral triangle).

2. Pythagorean Theorem:
   Using the Pythagorean theorem for one of these right triangles:
   $\left(\frac{a}{2}\right)^2 + h^2 = a^2$

3. Substitute $h = 10 \, \text{cm}$:
   Substitute $h = 10$ into the equation:
   $\left(\frac{a}{2}\right)^2 + 10^2 = a^2$

4. Simplify:
   Expand $\left(\frac{a}{2}\right)^2$:
   $\frac{a^2}{4} + 100 = a^2$

5. Eliminate the Fraction:
   Multiply through by 4 to eliminate the fraction:
   $a^2 + 400 = 4a^2$

6. Rearrange Terms:
   Simplify to isolate $a^2$:
   $400 = 3a^2$

7. Solve for $a^2$:
   $a^2 = \frac{400}{3}$

8. Solve for $a$:
   Take the square root of both sides:
   $a = \sqrt{\frac{400}{3}} = \frac{\sqrt{400}}{\sqrt{3}} = \frac{20}{\sqrt{3}} = \frac{20\sqrt{3}}{3} \, \text{cm}$

Final Answer:

The side length of the equilateral triangle is:
$a = \frac{20\sqrt{3}}{3} \, \text{cm} \approx 11.55 \, \text{cm}$

Would you like additional clarification or further details?

Related Questions:

1. How can the Pythagorean theorem be applied to other shapes, such as squares or trapezoids?
2. What is the relationship between the side length and area of an equilateral triangle?
3. How can the height of an equilateral triangle be derived directly from its side length?
4. Could this calculation be simplified by leaving the answer in radical form?
5. What are some real-life applications of equilateral triangles in design or architecture?

Tip:

Always check if simplifying or approximating values (like $\sqrt{3}$) is necessary, depending on whether the result will be used for theoretical or practical purposes.