Hypotenuse Length: Find X With Pythagorean Theorem

by Alex Johnson 51 views

Let's dive into how to find the length of the hypotenuse of a right triangle using the Pythagorean theorem. This is a fundamental concept in mathematics, especially in geometry and trigonometry, and understanding it can unlock solutions to many practical problems. We'll break down the theorem, explain its components, and apply it to a specific example. So, grab your thinking cap, and let's get started!

Understanding the Pythagorean Theorem

The Pythagorean Theorem is a cornerstone of geometry, providing a fundamental relationship between the sides of a right triangle. A right triangle, as you might recall, is a triangle that contains one angle that measures exactly 90 degrees—a right angle. The side opposite the right angle is called the hypotenuse, and it's always the longest side of the triangle. The other two sides are called legs.

The theorem itself is expressed as a simple equation:

a2+b2=c2{a^2 + b^2 = c^2}

Where:

  • a{ a } and b{ b } are the lengths of the legs of the right triangle.
  • c{ c } is the length of the hypotenuse.

In simpler terms, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. This relationship holds true for all right triangles, regardless of their size or orientation. Understanding this theorem is crucial not only for academic purposes but also for various real-world applications, such as construction, navigation, and even art.

The beauty of the Pythagorean Theorem lies in its ability to allow us to find the length of one side of a right triangle if we know the lengths of the other two sides. This makes it an incredibly powerful tool in problem-solving. Whether you're calculating the distance a baseball travels when thrown from home plate to second base, or determining the proper length of a support beam in construction, the Pythagorean Theorem provides a reliable method for finding these unknown lengths.

Furthermore, the Pythagorean Theorem serves as a foundation for many other concepts in mathematics. Trigonometric functions, such as sine, cosine, and tangent, are all defined in relation to the sides of a right triangle, and their values are heavily dependent on the relationships described by the theorem. In calculus, the theorem is used to find distances in coordinate systems and to define various geometric properties. Its influence extends even into more advanced fields like physics and engineering, where it is used to analyze forces, calculate trajectories, and design structures.

To truly grasp the theorem, it’s essential to practice applying it to different problems. This will help you become comfortable with identifying the legs and hypotenuse of a right triangle, setting up the equation correctly, and solving for the unknown side. With practice, you’ll find that the Pythagorean Theorem becomes an intuitive and indispensable tool in your mathematical toolkit.

Applying the Theorem to Our Problem

In our specific problem, we have a right triangle with legs of length 16 and 63. We are asked to find the length of the hypotenuse, which we'll call x{ x }. According to the Pythagorean Theorem, the sum of the squares of the legs is equal to the square of the hypotenuse. Therefore, we can set up the equation as follows:

162+632=x2{16^2 + 63^2 = x^2}

This equation directly applies the Pythagorean Theorem to the given values. The square of 16 (162{ 16^2 }) represents the square of the length of one leg, and the square of 63 (632{ 63^2 }) represents the square of the length of the other leg. The sum of these two squares is equal to the square of the length of the hypotenuse, x2{ x^2 }.

Now, let's analyze why the other options are incorrect:

  • Option B: 16+63=x{ 16 + 63 = x } suggests that the length of the hypotenuse is simply the sum of the lengths of the legs. This is incorrect because the Pythagorean Theorem involves squaring the lengths of the legs before adding them.
  • Option C: (16+63)2=x2{ (16 + 63)^2 = x^2 } suggests that we first add the lengths of the legs and then square the result to find the square of the hypotenuse. This is also incorrect because the theorem requires us to square the lengths of the legs individually before adding them.
  • Option D: 162+63=x{ 16^2 + 63 = x } is incorrect because it only squares one of the legs and then adds the other leg to it, which does not follow the Pythagorean Theorem.

Therefore, the correct equation to find the length of the hypotenuse x{ x } is:

162+632=x2{16^2 + 63^2 = x^2}

This equation accurately represents the relationship between the sides of the right triangle as defined by the Pythagorean Theorem. It sets the foundation for solving for x{ x } by first calculating the squares of the legs, summing them, and then taking the square root of the result to find the length of the hypotenuse. Understanding why this equation is correct and the others are not reinforces the fundamental principles of the theorem and ensures a solid grasp of its application.

Solving for x

Now that we've identified the correct equation, 162+632=x2{ 16^2 + 63^2 = x^2 }, let's solve for x{ x } to find the actual length of the hypotenuse.

First, we need to calculate the squares of 16 and 63:

162=256{16^2 = 256}

632=3969{63^2 = 3969}

Next, we add these two values together:

256+3969=4225{256 + 3969 = 4225}

So, our equation now looks like this:

x2=4225{x^2 = 4225}

To find x{ x }, we need to take the square root of both sides of the equation:

x=4225{x = \sqrt{4225}}

x=65{x = 65}

Therefore, the length of the hypotenuse x{ x } is 65. This means that in a right triangle with legs of length 16 and 63, the longest side, or the hypotenuse, measures 65 units. This calculation demonstrates the practical application of the Pythagorean Theorem in determining unknown lengths in right triangles.

By following these steps, you can confidently solve for the length of the hypotenuse in any right triangle, provided you know the lengths of the other two sides. Remember to square the lengths of the legs, add the results, and then take the square root to find the hypotenuse. This process not only provides the answer but also reinforces your understanding of the theorem and its usefulness in various mathematical and real-world contexts.

Conclusion

In summary, the equation that can be used to find x{ x }, the length of the hypotenuse of the right triangle with legs of length 16 and 63, is:

162+632=x2{16^2 + 63^2 = x^2}

This equation is a direct application of the Pythagorean Theorem, which states that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Understanding and applying this theorem is crucial for solving various problems in geometry, trigonometry, and other areas of mathematics. Remember to always identify the legs and hypotenuse correctly and set up the equation accordingly to ensure accurate results.

For further exploration of the Pythagorean Theorem and its applications, consider visiting Khan Academy for comprehensive lessons and practice exercises.