Place the numbers in the correct slots to describe the sides of the special right triangles. √√3 2 √2 al ....... The hypotenuse of a 45-45-90 right triangle is The hypotenuse of a 30-60-90 right triangle is The longer leg of a 30-60-90 right triangle is 12/0 times longer than its leg. times longer than its shortest leg. B times longer than its shortest leg.
Place the numbers in the correct slots to describe the sides of the special right triangles. √√3 2 √2 al ....... The hypotenuse of a 45-45-90 right triangle is The hypotenuse of a 30-60-90 right triangle is The longer leg of a 30-60-90 right triangle is 12/0 times longer than its leg. times longer than its shortest leg. B times longer than its shortest leg.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.6: Inequalities
Problem 78E
Related questions
Question
- 2
- 
- 1/2
**Descriptions:**
1. **The hypotenuse of a 45-45-90 right triangle is _______ times longer than its leg.**
- **Slot A: √2**
2. **The hypotenuse of a 30-60-90 right triangle is _______ times longer than its shortest leg.**
- **Slot B: 2**
3. **The longer leg of a 30-60-90 right triangle is ______ times longer than its shortest leg.**
- **Slot C: √3**
---
### Detailed Explanation:
In **45°-45°-90° triangles**, the lengths of the sides follow the ratio of 1:1:√2. This means:
- Each leg is of equal length.
- The hypotenuse (the side opposite the right angle) is √2 times longer than either leg.
In **30°-60°-90° triangles**, the lengths follow the ratio of 1:√3:2. This means:
- The shortest leg (opposite the 30° angle) is half the length of the hypotenuse.
- The hypotenuse is twice the length of the shortest leg.
- The longer leg (opposite the 60° angle) is √3 times longer than the shortest leg.
Understanding these relationships helps students solve problems involving these special right triangles more efficiently. By recognizing the patterns, students can quickly compute unknown side lengths given one side length.
For any further geometry learning resources or interactive activities, feel free to explore our educational content!
**Note:** The screenshot also shows placeholders labeled A, B, and C where students are expected to drag and drop the correct numbers corresponding to the side length ratios.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0890c2ce-3692-4718-a578-4f9466fde9bf%2Ff9fb3af7-d0f5-4176-901b-da23fcc3dbcc%2Foxexnqk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Special Right Triangles**
Special right triangles are right triangles with some "special" ratios of their sides. These include the 30°-60°-90° triangle and the 45°-45°-90° triangle. This section explores these triangles and the mathematical relationships between their sides.
**Fill in the numbers in the correct slots to describe the sides of special right triangles:**
**Given Numbers:**
- 
- 2
- 
- 1/2
**Descriptions:**
1. **The hypotenuse of a 45-45-90 right triangle is _______ times longer than its leg.**
- **Slot A: √2**
2. **The hypotenuse of a 30-60-90 right triangle is _______ times longer than its shortest leg.**
- **Slot B: 2**
3. **The longer leg of a 30-60-90 right triangle is ______ times longer than its shortest leg.**
- **Slot C: √3**
---
### Detailed Explanation:
In **45°-45°-90° triangles**, the lengths of the sides follow the ratio of 1:1:√2. This means:
- Each leg is of equal length.
- The hypotenuse (the side opposite the right angle) is √2 times longer than either leg.
In **30°-60°-90° triangles**, the lengths follow the ratio of 1:√3:2. This means:
- The shortest leg (opposite the 30° angle) is half the length of the hypotenuse.
- The hypotenuse is twice the length of the shortest leg.
- The longer leg (opposite the 60° angle) is √3 times longer than the shortest leg.
Understanding these relationships helps students solve problems involving these special right triangles more efficiently. By recognizing the patterns, students can quickly compute unknown side lengths given one side length.
For any further geometry learning resources or interactive activities, feel free to explore our educational content!
**Note:** The screenshot also shows placeholders labeled A, B, and C where students are expected to drag and drop the correct numbers corresponding to the side length ratios.
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