Assignment 3.4 1. What are the 2 special right triangles? (9. Give its properties.

### **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.