Solved Problems: Identifying Right Triangles, Legs, and Hypotenuse

📚 Identifying Right Triangles

A right triangle is a triangle that contains one angle of exactly 90 degrees. This angle is often marked with a small square. The sides of a right triangle have specific names, which are important to understand for various calculations.

📐 Key Components

• 🔍 Right Angle: The 90-degree angle, often indicated by a small square at the vertex.
• 🦵 Legs: The two sides that form the right angle. These are also known as the 'adjacent' and 'opposite' sides relative to a non-right angle in the triangle.
• 🎢 Hypotenuse: The side opposite the right angle; it's always the longest side of the right triangle.

📜 History and Background

The study of right triangles dates back to ancient civilizations, with significant contributions from the Egyptians, Babylonians, and Greeks. The most famous theorem associated with right triangles is the Pythagorean Theorem, which states a fundamental relationship between the lengths of the sides.

➕ The Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the other two sides ($a$ and $b$). Mathematically, this is represented as:

$a^2 + b^2 = c^2$

💡 Steps to Identify Legs and Hypotenuse

• 👁️ Locate the Right Angle: First, identify the 90-degree angle in the triangle. It's usually marked with a small square.
• 👉 Identify the Hypotenuse: The side opposite the right angle is the hypotenuse. It's always the longest side.
• ✍️ Identify the Legs: The two sides that form the right angle are the legs.

🌍 Real-World Examples

Example 1: Imagine a ladder leaning against a wall, forming a right triangle with the ground. The ladder is the hypotenuse, the wall and the ground are the legs.

Example 2: In construction, builders use right triangles to ensure that walls are perfectly vertical and floors are level.

➗ Solved Problems

Problem 1: A right triangle has legs of length 3 and 4. Find the length of the hypotenuse.

Solution: Using the Pythagorean Theorem:

$3^2 + 4^2 = c^2$

$9 + 16 = c^2$

$25 = c^2$

$c = 5$

The hypotenuse has a length of 5.

Problem 2: A right triangle has a hypotenuse of length 13 and one leg of length 5. Find the length of the other leg.

Solution: Using the Pythagorean Theorem:

$5^2 + b^2 = 13^2$

$25 + b^2 = 169$

$b^2 = 144$

$b = 12$

The other leg has a length of 12.

📝 Practice Quiz

Determine if the following sets of side lengths could form a right triangle. If so, identify the hypotenuse.

1. 3, 4, 5
2. 5, 12, 13
3. 8, 15, 16
4. 7, 24, 25
5. 6, 8, 10
6. 9, 12, 15
7. 10, 24, 26

Answers:

1. Yes, hypotenuse = 5
2. Yes, hypotenuse = 13
3. No
4. Yes, hypotenuse = 25
5. Yes, hypotenuse = 10
6. Yes, hypotenuse = 15
7. Yes, hypotenuse = 26

🎯 Conclusion

Understanding right triangles and their components is crucial in mathematics and various real-world applications. By mastering the identification of legs and the hypotenuse, and by applying the Pythagorean Theorem, you can solve a wide range of problems related to geometry and trigonometry.