Solving for the Hypotenuse of a Right Triangle with Given Perimeter and Area

When dealing with a right triangle, it is sometimes necessary to determine the length of the hypotenuse using the perimeter and area of the triangle. This article will guide you through solving this exact problem. We will explore how to use the given information about the perimeter and area to find the hypotenuse of a right triangle using algebraic manipulations and the Pythagorean Theorem.

Understanding the Problem and Key Elements

A right triangle has a perimeter of 32 cm and an area of 20 cm2. Let's denote the hypotenuse as c, and the other two sides as a and b. The goal is to find the length of the hypotenuse c.

Exploring the Relationship Between Perimeter and Area

The perimeter of the triangle is given by the sum of its sides:

[a b c 32]

The area of the triangle is given by the formula:

[frac{1}{2}ab 20 Rightarrow ab 40]

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:

[a^2 b^2 c^2]

Derivation of the Hypotenuse Length

We can use the equations derived from the given information to find c:

From the area equation: Substitute ab 40 in the Pythagorean Theorem equation: [a^2 b^2 c^2] [a^2 b^2 40 c^2 - 2ab] [a^2 b^2 40 c^2 - 80] [a^2 b^2 c^2 - 40] Using the perimeter equation, we can express a b as: [a b 32 - c] Square this equation: [(a b)^2 (32 - c)^2] [a^2 2ab b^2 1024 - 64c c^2] Now replace (a^2 b^2) with (c^2 - 40): [c^2 - 40 80 1024 - 64c c^2] [c^2 40 1024 - 64c c^2] Subtract (c^2) from both sides: [40 1024 - 64c] Subtract 1024 from both sides: [-984 -64c] Divide both sides by -64: [c frac{984}{64} 15.375] Thus, the length of the hypotenuse c is approximately 14.75 cm.

Conclusion

Using the given perimeter and area, we can determine the length of the hypotenuse of a right triangle. The hypotenuse, in this case, is approximately 14.75 cm. This solution utilizes the Pythagorean Theorem and algebraic manipulation to arrive at the correct answer. Understanding these concepts and their applications is crucial in solving complex geometric problems.

Key Takeaways

Pythagorean Theorem: In a right triangle, (a^2 b^2 c^2). Perimeter: The sum of the lengths of all sides. The space enclosed by the triangle, calculated as (frac{1}{2}ab).

For more information on geometry and related problems, refer to reputable mathematical resources and online forums. Understanding these fundamental principles can be incredibly valuable in various fields and applications, from engineering to physics and beyond.