Solving for the Dimensions of a Rectangle Using the Pythagorean Theorem

Understanding the relationship between the sides and the diagonal of a rectangle is crucial in geometry. This article will demonstrate how to solve for the dimensions of a rectangle when given the length of the diagonal and a specific relationship between its sides, using the Pythagorean Theorem.

Problem Context

Consider a rectangle where one side is 2 meters longer than the other, and the diagonal measures 11 meters. This problem can be solved using the Pythagorean Theorem, which states that in a right-angled triangle (which a rectangle's diagonal forms with its sides), the square of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the other two sides.

Step-by-Step Solution

Let's denote the shorter side of the rectangle as n meters. Therefore, the longer side would be n 2 meters. The diagonal of the rectangle, which is 11 meters, forms a right-angled triangle with the sides.

Application of the Pythagorean Theorem

The Pythagorean Theorem can be written as:

n^2 (n 2)^2 11^2

Expanding and simplifying this equation:

n^2 n^2 4n 4 121

Merging like terms:

2n^2 4n - 117 0

This is a quadratic equation, which can be solved using the quadratic formula:

n frac{-b pm sqrt{b^2 - 4ac}}{2a}

Here, a 2, b 4, and c -117.

Plugging in these values:

n frac{-4 pm sqrt{4^2 - 4 cdot 2 cdot (-117)}}{2 cdot 2}

n frac{-4 pm sqrt{16 936}}{4}

n frac{-4 pm sqrt{952}}{4}

n frac{-4 pm 30.85}{4}

This gives us two solutions:

n frac{26.85}{4} 6.71

n frac{-34.85}{4} -8.71

Since dimensions cannot be negative, we discard n -8.71 and accept n 6.71.

Therefore, the sides of the rectangle are 6.71 meters and 8.71 meters (since n 2 8.71).

Alternative Solution

Another approach involves directly solving the equation:

2w^2 4w - 121 0

Using the quadratic formula again, we find:

w frac{-4 pm sqrt{4^2 - 4 cdot 2 cdot (-121)}}{2 cdot 2}

w frac{-4 pm sqrt{16 968}}{4}

w frac{-4 pm sqrt{984}}{4}

w frac{-4 pm 31.37}{4}

This gives us two solutions:

w frac{27.37}{4} 6.84

w frac{-35.37}{4} -8.84

Discarding the negative solution, we have w 6.84. So, the sides are approximately 6.84 meters and 8.84 meters.

Conclusion

By applying the Pythagorean Theorem and solving the resulting quadratic equation, we can accurately determine the dimensions of a rectangle given its diagonal and the relationship between its sides. This problem illustrates the practical use of the theorem in solving real-world dimensional challenges.

Conclusion

In conclusion, understanding and applying the Pythagorean Theorem is essential for solving many geometric and practical problems. Whether using the original equation or the simplified version, the method remains the same, and the solution will be accurate as long as the values are correctly calculated.

Key Takeaways

The Pythagorean Theorem: a^2 b^2 c^2 Quadratic Equation: n frac{-b pm sqrt{b^2 - 4ac}}{2a} Real-world applications: solving for dimensions of rectangles

Keywords: Pythagorean Theorem, Rectangle Diagonal, Rectangular Dimensions