Introduction

The simplicity and elegance of a right triangle make it a cornerstone in geometry, often serving as a vital tool in trigonometry, architecture, and engineering. This article delves into a specific problem: determining the hypotenuse of a right triangle with given leg lengths and the perimeter. We will explore the application of the Pythagorean theorem and the principles of perimeter calculation to arrive at the solution. By the end of this article, you will have a comprehensive understanding of how to solve such problems and a deeper appreciation for the interplay between basic geometric principles.

Problem Statement

Given a right triangle with legs of length 6 and b, and a hypotenuse of length c, where the perimeter of the triangle is 18, determine the length of the hypotenuse c.

Step-by-Step Solution

Step 1: Understanding the Perimeter

The perimeter of a triangle is the sum of the lengths of its sides. For the given right triangle, the perimeter is:

a b c 18

Given that one leg a 6, the equation becomes:

6 b c 18

Simplifying further, we get:

b c 12 ... Equation 1

Step 2: Applying the Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

a2 b2 c2

Substituting a 6, we have:

62 b2 c2

This simplifies to:

36 b2 c2 ... Equation 2

Step 3: Substituting and Solving

From Equation 1, we can express c in terms of b as follows:

c 12 - b

Substituting c 12 - b into Equation 2 gives:

36 b2 12 - b2

Expanding and rearranging:

36 b2 144 - 24b b2

Subtracting b2 from both sides:

36 144 - 24b

Rearranging the equation to solve for b:

24b 144 - 36

24b 108

Dividing both sides by 24:

b frac{108}{24} 4.5

Step 4: Finding c

Substituting b 4.5 back into Equation 1 gives:

c 12 - b 12 - 4.5 7.5

Therefore, the length of the hypotenuse c is:

Answer: c 7.5

Additional Insights

Insight 1: Perimeter and Pythagorean Theorem Combined

The problem effectively demonstrates how the Pythagorean theorem and the perimeter formula can be used in tandem to solve for unknown values in a right triangle. This is a practical application of these fundamental geometric principles in real-world scenarios.

Insight 2: System of Equations

The problem is essentially a system of linear and quadratic equations, which are common in geometric problems. Mastering these methods is crucial for solving more complex geometric scenarios in various fields, including physics and engineering.

Insight 3: Numerical Methods

The solution involves numerical methods, which are widely used in computational geometry and numerical analysis. Understanding these methods is essential for anyone dealing with mathematical modeling in applied sciences.