https://file.aiquickdraw.com/mathbot/file/17290043607853vl483xi.jpg

This question concerns the properties of a **30-60-90 triangle**, a special type of right triangle. In a 30-60-90 triangle, the sides follow these relationships:

- The shorter leg (opposite the 30° angle) is \( x \).
- The longer leg (opposite the 60° angle) is \( x\sqrt{3} \).
- The hypotenuse is \( 2x \).

Let's evaluate the given options:

- **A. The hypotenuse is twice as long as the shorter leg.**
  - This is **true** because the hypotenuse in a 30-60-90 triangle is \( 2x \), where \( x \) is the shorter leg.

- **B. The longer leg is \( \sqrt{3} \) times as long as the shorter leg.**
  - This is **true** because the longer leg is \( x\sqrt{3} \), where \( x \) is the shorter leg.

- **C. The longer leg is twice as long as the shorter leg.**
  - This is **false**. The longer leg is \( \sqrt{3} \) times the shorter leg, not twice as long.

- **D. The hypotenuse is \( \sqrt{3} \) times as long as the longer leg.**
  - This is **false**. The hypotenuse is twice the length of the shorter leg, not \( \sqrt{3} \) times the longer leg.

- **E. The hypotenuse is twice as long as the longer leg.**
  - This is **false**. The hypotenuse is longer than the longer leg, but it is not exactly twice as long. It's only twice the shorter leg.

- **F. The hypotenuse is \( \sqrt{3} \) times as long as the shorter leg.**
  - This is **false**. The hypotenuse is \( 2x \), while \( x\sqrt{3} \) is the longer leg.

### Correct answers: **A** and **B**.

Would you like further details or have any additional questions?

Here are some related questions:
1. How can you derive the side ratios of a 30-60-90 triangle using the Pythagorean theorem?
2. Why does the longer leg in a 30-60-90 triangle have a \( \sqrt{3} \) relationship with the shorter leg?
3. How can you apply the side ratios of a 30-60-90 triangle to solve real-world problems?
4. Can the properties of a 30-60-90 triangle be applied to non-right triangles?
5. How do the properties of a 45-45-90 triangle differ from those of a 30-60-90 triangle?

**Tip**: Memorizing the side ratios for special triangles like the 30-60-90 triangle can significantly speed up solving geometry problems.

Which of the following are true statements about a 30-60-90 triangle?

Understand the side ratios of a 30-60-90 triangle and determine which statements are true. Learn the relationships between the shorter leg, longer leg, and hypotenuse, following the formulas for this special right triangle. Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation