https://file.aiquickdraw.com/mathbot/file/1730485421353w9pn9ij5.jpg

To solve for the length of side \( x \) in simplest radical form, we need to analyze the structure of this equilateral triangle, where one altitude is drawn, creating two 30-60-90 right triangles.

### Solution Steps

1. **Understand the Properties of a 30-60-90 Triangle**:
   In a 30-60-90 triangle, the ratio of the lengths of the sides opposite the 30°, 60°, and 90° angles are \( 1 : \sqrt{3} : 2 \). Here:
   - The shortest side (opposite the 30° angle) is half of the hypotenuse.
   - The side opposite the 60° angle is \( \sqrt{3} \) times the shortest side.

2. **Identify the Known Side**:
   The hypotenuse of the right triangle in the diagram is 11, which is the same as the side of the equilateral triangle. Therefore, the side length of the equilateral triangle is 11.

3. **Calculate the Length of \( x \)**:
   Since \( x \) is the shorter leg of the 30-60-90 triangle (opposite the 30° angle), it is half of the hypotenuse.
   \[
   x = \frac{11}{2}
   \]

4. **Conclusion**:
   Thus, the length of \( x \) is:
   \[
   x = \frac{11}{2}
   \]

This is the simplest radical form with a rational denominator for \( x \).

Would you like further details or have any questions?

---

### Related Questions
1. How would you find the height of an equilateral triangle if you know the side length?
2. What is the area of an equilateral triangle with side length 11?
3. How do the properties of 30-60-90 triangles simplify calculations in equilateral triangles?
4. If \( x \) were given, how could you determine the side length of the equilateral triangle?
5. Can you explain why drawing an altitude in an equilateral triangle creates two 30-60-90 triangles?

### Tip
Remember that an altitude in an equilateral triangle always bisects the opposite side and creates two 30-60-90 triangles, a useful property for many geometric problems.

The triangle below is equilateral. Find the length of side x in simplest radical form with a rational denominator.

Solve for side x in an equilateral triangle with a hypotenuse of 11, using 30-60-90 triangle properties. Includes step-by-step calculations for finding the shortest side in simplest radical form with a rational denominator, ideal for geometry students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation