Right Triangle Side Length: Step-by-Step Solution

by ADMIN 50 views

Hey guys! Let's dive into a classic geometry problem involving right triangles. We're given some info and need to find the length of a missing side. It's like a puzzle, and who doesn't love puzzles? So grab your thinking caps, and let's get started!

Understanding the Problem

Our mission, should we choose to accept it, is to find the length of the third side of a right triangle. We know it's a right triangle because angle B is a sweet 90 degrees. The longest side, the hypotenuse, is 61 cm, and the shortest side is 11 cm. We have to use the Pythagorean Theorem magic to discover the length of the remaining side. The Pythagorean Theorem is a fundamental concept in geometry that describes the relationship between the sides of a right triangle. It states that the sum of the squares of the two shorter sides (legs) of a right triangle is equal to the square of the longest side (hypotenuse). Mathematically, it's expressed as a² + b² = c², where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse. Understanding and applying this theorem is key to solving many geometry problems involving right triangles. The hypotenuse is always opposite the right angle and is the longest side of the triangle. In our problem, we know the length of the hypotenuse (c = 61 cm) and the length of one leg (let's say a = 11 cm). We need to find the length of the other leg, which we'll call 'b'. By plugging the known values into the Pythagorean Theorem, we can set up an equation to solve for 'b'. This involves some algebraic manipulation, but don't worry, we'll take it step by step. The Pythagorean Theorem is not just a formula; it's a powerful tool that helps us understand the relationships between the sides of a right triangle. It has numerous applications in various fields, including architecture, engineering, and navigation. Mastering this theorem is essential for anyone studying geometry or related subjects. So, let's get ready to apply this theorem to solve our problem and find the missing side of the right triangle.

Applying the Pythagorean Theorem

The Pythagorean Theorem is our trusty tool here. Remember, it states: a² + b² = c², where 'c' is the hypotenuse (the longest side) and 'a' and 'b' are the other two sides (legs). We know c = 61 cm and let's say a = 11 cm. We need to find 'b'. Let's plug in the values we know: 11² + b² = 61². Now, let's do some math. 11² = 121, and 61² = 3721. So our equation becomes: 121 + b² = 3721. To isolate b², we need to subtract 121 from both sides of the equation: b² = 3721 - 121, which simplifies to b² = 3600. Now, to find 'b', we need to take the square root of both sides: b = √3600. The square root of 3600 is 60. Therefore, b = 60 cm. This means the length of the third side of the right triangle is 60 centimeters. We've successfully used the Pythagorean Theorem to find the missing side. This theorem is a fundamental concept in geometry and has numerous applications in various fields. Understanding and applying it correctly is essential for solving problems involving right triangles. Remember, the hypotenuse is always the longest side and is opposite the right angle. By plugging in the known values and using algebraic manipulation, we can find the length of any missing side of a right triangle. So, next time you encounter a right triangle problem, don't panic! Just remember the Pythagorean Theorem, and you'll be able to solve it with ease.

Solution

Therefore, the length of the third side is 60 cm. That wasn't so bad, right? The key was understanding and applying the Pythagorean Theorem. Always remember to identify the hypotenuse and the legs correctly before plugging the values into the formula. This ensures that you get the correct answer. Geometry problems often seem daunting at first, but with a solid understanding of the fundamental concepts and a bit of practice, you can solve them with confidence. So keep practicing, and you'll become a geometry whiz in no time! Remember to always double-check your work to ensure that your answer makes sense in the context of the problem. In this case, we found that the third side is 60 cm, which is less than the hypotenuse (61 cm) but greater than the shortest side (11 cm), which makes sense. This helps us verify that our answer is reasonable. So, keep learning, keep practicing, and keep having fun with math!

Practice Problems

Why not test your understanding? Try solving these similar problems:

  1. A right triangle has legs of length 5 cm and 12 cm. What is the length of the hypotenuse?
  2. The hypotenuse of a right triangle is 25 cm, and one leg is 7 cm. Find the length of the other leg.

Solving these problems will solidify your understanding of the Pythagorean Theorem and improve your problem-solving skills. Remember to follow the same steps we used in the example problem: identify the hypotenuse and the legs, plug in the known values into the formula, and solve for the unknown side. Don't be afraid to draw diagrams to help visualize the problem. This can make it easier to understand the relationships between the sides of the right triangle. And most importantly, don't give up! If you get stuck, review the steps we took in the example problem and try again. With practice, you'll become more confident and proficient in solving right triangle problems.

Conclusion

In summary, we successfully found the missing side of a right triangle using the Pythagorean Theorem. Remember, a² + b² = c². Keep practicing, and you'll be a pro in no time! Geometry is a fascinating branch of mathematics that has numerous applications in the real world. From architecture to engineering to computer graphics, geometry plays a crucial role in many fields. By mastering the fundamental concepts of geometry, you'll be well-equipped to tackle a wide range of problems and challenges. So keep exploring, keep learning, and keep having fun with geometry! Remember, math is not just about numbers and formulas; it's about problem-solving, critical thinking, and logical reasoning. These skills are valuable in all aspects of life, so keep honing them and you'll be successful in whatever you do. And always remember to approach problems with a positive attitude and a willingness to learn. With hard work and dedication, you can achieve anything you set your mind to. So go out there and conquer the world of geometry!