Answer
The two special right triangles are the 45-45-90 triangle and the 30-60-90 triangle. They have fixed angles and consistent side length ratios that make them useful in geometry.
Solution
### **Assignment 3.4**
#### **1. What are the 2 Special Right Triangles?**
The two special right triangles are:
1. **45-45-90 Triangle**
2. **30-60-90 Triangle**
These triangles are termed "special" because their angles are fixed, and their side length ratios are consistent, making them particularly useful in various geometric calculations and proofs.
---
#### **9. Give Their Properties**
##### **1. 45-45-90 Triangle**
- **Angles:** This triangle has two angles of 45 degrees and one right angle (90 degrees).
- **Side Length Ratios:** The sides opposite the 45-degree angles are equal in length, making the triangle isosceles. The ratio of the lengths of the sides is:
\[
1 : 1 : \sqrt{2}
\]
- **Legs:** Both legs (the sides opposite the 45-degree angles) are of equal length. If each leg is of length \( x \), then:
\[
\text{Leg}_1 = x \quad \text{and} \quad \text{Leg}_2 = x
\]
- **Hypotenuse:** The side opposite the right angle (hypotenuse) is:
\[
\text{Hypotenuse} = x\sqrt{2}
\]
- **Properties:**
- **Isosceles Nature:** Due to the equal legs, the triangle is isosceles.
- **Simplified Calculations:** Knowing the side ratios allows for easy calculation of unknown sides when one side is given.
- **Applications:** Often used in problems involving symmetry, bisectors, and in constructing squares and other geometric shapes.
##### **2. 30-60-90 Triangle**
- **Angles:** This triangle has one angle of 30 degrees, one of 60 degrees, and one right angle (90 degrees).
- **Side Length Ratios:** The sides are in the ratio:
\[
1 : \sqrt{3} : 2
\]
- **Shortest Side (Opposite 30°):** Let's denote this side as \( x \).
- **Longer Leg (Opposite 60°):**
\[
\text{Longer Leg} = x\sqrt{3}
\]
- **Hypotenuse (Opposite 90°):**
\[
\text{Hypotenuse} = 2x
\]
- **Properties:**
- **Unique Angle Measures:** The distinct angles lead to the specific side ratios, which are essential in trigonometry and geometry.
- **Scalability:** Knowing one side allows for the determination of the other two sides using the fixed ratio.
- **Applications:** Commonly used in problems involving equilateral triangles (by bisecting them), trigonometric identities, and real-world applications like architecture and engineering designs.
---
### **Summary of Side Ratios**
| Triangle Type | Angles | Side Ratios |
|---------------|-----------------|---------------------|
| 45-45-90 | 45°, 45°, 90° | \( 1 : 1 : \sqrt{2} \) |
| 30-60-90 | 30°, 60°, 90° | \( 1 : \sqrt{3} : 2 \) |
### **Visual Representation**
**45-45-90 Triangle:**
```
/|
/ |
x / | x
/___|
x√2
```
**30-60-90 Triangle:**
```
/
/|
/ |
x√3/ | 2x
/___|
x
```
Understanding these special right triangles simplifies solving many geometric problems, especially those involving trigonometric functions and the Pythagorean theorem.
Reviewed and approved by the UpStudy tutoring team
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