Calculating Hypotenuse Length Using The Pythagorean Theorem

In the realm of mathematics, particularly geometry, the Pythagorean Theorem stands as a cornerstone principle for understanding the relationships within right-angled triangles. This theorem, named after the ancient Greek mathematician Pythagoras, provides a fundamental equation that connects the lengths of the sides of a right triangle. In this article, we will delve into the application of the Pythagorean Theorem to determine the length of the hypotenuse, which is the side opposite the right angle. We will explore different scenarios and demonstrate how to formulate the correct equation to solve for the hypotenuse. Let's consider the given scenarios:

Understanding the Pythagorean Theorem

Before we dive into the specific equations, it's essential to have a firm grasp of the Pythagorean Theorem itself. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs). Mathematically, this is expressed as:

a² + b² = c²

Where:

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

This theorem is a powerful tool for solving various geometric problems, particularly those involving right triangles. It allows us to find missing side lengths when we know the lengths of the other two sides. To effectively apply the Pythagorean Theorem, it's crucial to correctly identify the hypotenuse and the legs of the right triangle. The hypotenuse is always the longest side and is opposite the right angle. The legs are the two sides that form the right angle.

Now, let's explore the given scenarios and see how we can apply this theorem to find the length of the hypotenuse.

Scenario 1: B² + 11² = c²

In this scenario, we are presented with the equation B² + 11² = c². This equation closely resembles the Pythagorean Theorem (a² + b² = c²). By comparing the two equations, we can infer that:

• B represents the length of one leg of the right triangle.
• 11 represents the length of the other leg of the right triangle.
• c represents the length of the hypotenuse.

This equation directly applies the Pythagorean Theorem to find the length of the hypotenuse (c). We are given the lengths of the two legs (B and 11), and we need to solve for the hypotenuse (c). To do this, we would first square the lengths of the legs, add them together, and then take the square root of the sum to find the value of c.

For example, if B were 5, the equation would become:

5² + 11² = c²

25 + 121 = c²

146 = c²

c = √146 ≈ 12.08

Therefore, the equation B² + 11² = c² is a valid equation that can be used to find the length of the hypotenuse (c) in a right triangle, given the lengths of the two legs (B and 11).

Scenario 2: 8² + c² = 11²

In this scenario, we are given the equation 8² + c² = 11². Again, this equation is related to the Pythagorean Theorem, but it's structured slightly differently. By comparing this equation to the standard form of the theorem (a² + b² = c²), we can see that:

• 8 represents the length of one leg of the right triangle.
• c represents the length of the other leg of the right triangle.
• 11 represents the length of the hypotenuse.

In this case, we are given the length of one leg (8) and the length of the hypotenuse (11), and we are asked to find the length of the other leg (c). To solve for c, we would need to rearrange the equation to isolate c² on one side. This can be done by subtracting 8² from both sides of the equation:

c² = 11² - 8²

Now, we can solve for c by taking the square root of the result.

For example:

c² = 121 - 64

c² = 57

c = √57 ≈ 7.55

Therefore, the equation 8² + c² = 11² is not used to find the length of the hypotenuse. Instead, it is used to find the length of one of the legs of the right triangle when the length of the other leg and the hypotenuse are known.

Conclusion

In conclusion, when seeking to determine the length of the hypotenuse in a right-angled triangle using the Pythagorean Theorem, the equation that directly applies is the one where the square of the hypotenuse is isolated on one side of the equation, and the sum of the squares of the legs is on the other side. In the given scenarios, the equation B² + 11² = c² is the correct equation to use to find the length of the hypotenuse, where c represents the hypotenuse, and B and 11 represent the lengths of the legs. The equation 8² + c² = 11², on the other hand, is used to find the length of one of the legs when the hypotenuse and the other leg are known. Understanding the structure and application of the Pythagorean Theorem is crucial for solving a wide range of geometric problems involving right triangles.

By mastering the Pythagorean Theorem, you gain a powerful tool for solving various problems related to right triangles. Remember to always identify the hypotenuse and legs correctly, and apply the theorem accordingly. Whether you are finding the hypotenuse or a leg, the Pythagorean Theorem provides a reliable method for determining the missing side lengths.