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We are given an equilateral triangle with a right triangle inside it. One leg of the right triangle is labeled as 6, and we are tasked with finding the hypotenuse \( x \), which represents the side length of the equilateral triangle.

### Approach:
1. **Equilateral Triangle Properties**: In an equilateral triangle, all sides are equal, and the angles are 60° each.
2. **Right Triangle**: We see that one of the sides of the equilateral triangle forms a right triangle inside. The given leg (6) is half of the height of the equilateral triangle, and the hypotenuse is the full side length \( x \).

Using trigonometric relationships or the properties of an equilateral triangle, the height \( h \) of an equilateral triangle is given by:
\[
h = \frac{\sqrt{3}}{2} \times s
\]
where \( s \) is the side length (in this case, \( x \)).

Here, the height is 6, so we can set up the equation:
\[
6 = \frac{\sqrt{3}}{2} \times x
\]

### Solving for \( x \):
\[
x = \frac{6 \times 2}{\sqrt{3}} = \frac{12}{\sqrt{3}}
\]

We rationalize the denominator by multiplying by \( \frac{\sqrt{3}}{\sqrt{3}} \):
\[
x = \frac{12 \sqrt{3}}{3} = 4\sqrt{3}
\]

Now, approximate \( \sqrt{3} \approx 1.732 \):
\[
x \approx 4 \times 1.732 = 6.928
\]

Rounding to the nearest tenth:
\[
x \approx 6.9
\]

Thus, the length of side \( x \) is approximately **6.9**.

Would you like further details or explanations?

Here are 5 related questions to consider:
1. How do you calculate the height of an equilateral triangle in general?
2. Can the Pythagorean theorem be applied here directly, and if so, how?
3. What are the properties of a 30-60-90 triangle, and how can they help in such problems?
4. How can you derive the area of an equilateral triangle using its side length?
5. What are some real-life applications of equilateral triangles?

**Tip**: Remember, equilateral triangles are highly symmetrical, and their properties simplify many geometry problems.

The triangle below is equilateral. Find the length of side x to the nearest tenth.

This solution demonstrates how to find the side length of an equilateral triangle by using trigonometric properties. The height of the triangle is provided as 6, and the solution involves using the formula for the height of an equilateral triangle. Suitable for students in Grades 9-11.

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