https://file.aiquickdraw.com/mathbot/file/1726702081932a0xwi8d9.jpg

The triangle in the diagram is equilateral, meaning all its sides are equal, and there’s a right triangle drawn inside it with a given side length of 9. To find \( x \), we will need to use properties of equilateral triangles and possibly apply the Pythagorean theorem.

### Step 1: Geometry of the Equilateral Triangle
An equilateral triangle has all angles measuring 60°. When we draw a perpendicular from one vertex to the opposite side, it bisects that side and the angle at the vertex. This forms two 30-60-90 right triangles.

In a 30-60-90 triangle:
- The hypotenuse is twice the length of the shorter leg (half of the equilateral triangle's side).
- The longer leg is \( \sqrt{3} \) times the shorter leg.

### Step 2: Analyze the Given Right Triangle
In the diagram:
- The side labeled \( 9 \) is the shorter leg of the 30-60-90 triangle.
- \( x \) is the hypotenuse of this right triangle, which corresponds to the side of the equilateral triangle.

Thus, we can use the relationship for a 30-60-90 triangle:
\[
\text{Hypotenuse} = 2 \times \text{Shorter Leg}
\]
\[
x = 2 \times 9 = 18
\]

Thus, the length of side \( x \) is \( 18 \) units.

Let me know if you'd like further details or clarifications.

Here are 5 related questions:
1. How do you find the height of an equilateral triangle given the side length?
2. What is the area of this equilateral triangle with side length \( x = 18 \)?
3. How is the Pythagorean theorem used in solving for unknown sides in right triangles?
4. Can the trigonometric ratios be used to solve for sides in this type of problem? How?
5. What are the properties of a 30-60-90 triangle?

**Tip:** In a 30-60-90 triangle, the ratios between the sides are always consistent, which makes it easy to solve for unknown side lengths.

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

Learn how to solve for the length of side x in an equilateral triangle using properties of 30-60-90 right triangles. This geometry problem involves applying the hypotenuse formula to find the unknown side length. Suitable for Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation