https://file.aiquickdraw.com/mathbot/file/1726550620026otvmg5c0.jpg

The problem shown in the image involves a **30°-60°-90° special right triangle**. For this type of triangle, the sides are in a specific ratio:

- The side opposite the 30° angle is half the hypotenuse.
- The side opposite the 60° angle is \( \frac{\sqrt{3}}{2} \) times the hypotenuse.

From the image:
- The hypotenuse is given as \( 27 \, \text{mm} \).
- We need to find the side \( m \), which is opposite the 60° angle.

### Step-by-step solution:

1. **Identify the relationship for the side opposite the 60° angle**:
   \[
   m = \frac{\sqrt{3}}{2} \times \text{hypotenuse}
   \]
2. **Substitute the value of the hypotenuse**:
   \[
   m = \frac{\sqrt{3}}{2} \times 27
   \]
3. **Simplify**:
   \[
   m = 27 \times \frac{\sqrt{3}}{2} = \frac{27\sqrt{3}}{2} \, \text{mm}
   \]

Thus, the length of \( m \) is \( \frac{27\sqrt{3}}{2} \, \text{mm} \), which is the simplest radical form.

Would you like further details or have any other questions?

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Here are 5 related questions:
1. How do you find the side opposite the 30° angle in a 30°-60°-90° triangle?
2. What is the general formula for the side lengths of a 30°-60°-90° triangle?
3. How would you calculate the area of this triangle?
4. What is the perimeter of this triangle, in terms of radicals?
5. Can this triangle be used to model any real-life applications?

**Tip**: Always remember the side ratios in special triangles like the 30°-60°-90°, as they make solving problems easier!

Find the length of side m in a 30°-60°-90° triangle where the hypotenuse is 27 mm. Write the answer in simplest radical form.

This page provides a step-by-step solution to find the side m in a 30°-60°-90° special right triangle, where the hypotenuse is 27 mm. We use the 30°-60°-90° triangle ratio and simplify the result to the simplest radical form. Perfect for high school geometry students in grades 9-11.

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