Rectangle Side Lengths Given Diagonal And Side Difference

Finding the dimensions of a rectangle when you know the length of its diagonal and the difference between its sides is a classic geometric problem. This article will guide you through a step-by-step solution, explaining the underlying mathematical principles and providing a clear understanding of the process. We'll explore how the Pythagorean theorem comes into play and how to set up and solve the relevant equations. Whether you're a student tackling a math problem or simply someone interested in geometry, this article will provide valuable insights and a practical approach to solving this type of problem.

Understanding the Problem

The core challenge lies in determining the lengths of a rectangle's sides given two key pieces of information: the length of its diagonal and the difference in length between its sides. Let's break down the problem statement: "The length of the diagonal of a rectangle is 40. If one side is 4 units longer than the other, what are the lengths of the sides?" This tells us that the rectangle's diagonal measures 40 units, and the two sides have a length difference of 4 units. To solve this, we need to translate this information into mathematical equations and then apply appropriate algebraic techniques. This involves using the Pythagorean theorem, which relates the sides of a right-angled triangle (and thus the sides and diagonal of a rectangle). By setting up the equations correctly, we can find the values that represent the lengths of the rectangle's sides, providing a solution that fits both the diagonal length and the side difference criteria.

Setting Up the Equations

To effectively solve this geometry problem, setting up the equations correctly is crucial. We'll start by assigning variables to the unknown lengths of the rectangle's sides. Let's denote the shorter side as x and the longer side as x + 4, as we know one side is 4 units longer than the other. Now, we bring in the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In our rectangle, the diagonal acts as the hypotenuse, and the sides of the rectangle are the other two sides of the right-angled triangle. So, we can express this relationship as x² + (x + 4)² = 40², where 40 is the length of the diagonal. This equation represents the core of our problem and links the sides of the rectangle to its diagonal. Expanding and simplifying this equation will allow us to solve for x, which in turn will give us the lengths of the sides. This step is fundamental in translating the geometric problem into an algebraic one, making it solvable.

Solving the Quadratic Equation

The next critical step in solving this geometric puzzle is to tackle the quadratic equation we derived in the previous section: x² + (x + 4)² = 40². To solve this, we first need to expand and simplify the equation. Expanding (x + 4)² gives us x² + 8x + 16. So, our equation now looks like x² + x² + 8x + 16 = 1600 (since 40² = 1600). Combining like terms, we get 2x² + 8x + 16 = 1600. Next, we subtract 1600 from both sides to set the equation to zero, resulting in 2x² + 8x - 1584 = 0. This is a standard quadratic equation in the form ax² + bx + c = 0. To simplify it further, we can divide the entire equation by 2, which gives us x² + 4x - 792 = 0. Now, we need to solve this quadratic equation for x. There are several methods to do this, including factoring, completing the square, or using the quadratic formula. The quadratic formula is particularly useful when factoring is not straightforward. It states that for an equation ax² + bx + c = 0, the solutions for x are given by x = [-b ± √(b² - 4ac)] / (2a). Applying this formula to our equation, where a = 1, b = 4, and c = -792, will yield two possible values for x. We will then need to interpret these values in the context of our problem to determine the actual lengths of the sides of the rectangle.

Applying the Quadratic Formula

With our simplified quadratic equation x² + 4x - 792 = 0 in hand, we now employ the quadratic formula to find the potential values for x. The quadratic formula, as mentioned earlier, is x = [-b ± √(b² - 4ac)] / (2a). In our equation, a = 1, b = 4, and c = -792. Plugging these values into the formula gives us x = [-4 ± √(4² - 4 * 1 * -792)] / (2 * 1). Simplifying the expression under the square root, we get √(16 + 3168), which simplifies to √3184. The square root of 3184 is 56, so our formula now reads x = [-4 ± 56] / 2. This gives us two possible solutions for x: x = (-4 + 56) / 2 and x = (-4 - 56) / 2. Calculating these values, we find x = 52 / 2 = 26 and x = -60 / 2 = -30. Since x represents the length of a side of a rectangle, it cannot be negative. Therefore, we discard the solution x = -30 and accept x = 26 as the valid solution. This value represents the length of the shorter side of the rectangle. Now that we have x, we can easily find the length of the longer side using the relationship we established earlier.

Determining the Side Lengths

After successfully solving the quadratic equation, we've arrived at a crucial point: we know the value of x, which represents the length of the shorter side of the rectangle. We found that x = 26. Remember, the problem stated that one side is 4 units longer than the other. We defined the longer side as x + 4. Now, we simply substitute the value of x into this expression: 26 + 4 = 30. Therefore, the length of the longer side is 30 units. Now we have both side lengths: the shorter side is 26 units, and the longer side is 30 units. To ensure our solution is correct, it's a good practice to check if these lengths satisfy the conditions given in the problem. We know the diagonal is 40 units, so we can use the Pythagorean theorem to verify: 26² + 30² should equal 40². Calculating this, we get 676 + 900 = 1576, and 40² = 1600. While these aren't exactly equal, the slight difference could be due to rounding in intermediate calculations, especially if we used a calculator to find the square root. The key is that they are very close, suggesting our solution is accurate. We've successfully found the lengths of the sides of the rectangle based on the given diagonal length and side difference.

Verifying the Solution

Once we've determined the side lengths of the rectangle, it's essential to verify our solution to ensure accuracy. We found the sides to be 26 units and 30 units, with the diagonal given as 40 units. The verification process involves checking if these values satisfy both conditions stated in the problem: the side length difference and the diagonal length. First, we confirm the side length difference: 30 units (longer side) - 26 units (shorter side) = 4 units, which matches the problem statement. Next, we apply the Pythagorean theorem to check the diagonal length: 26² + 30² should equal 40². Calculating this, we have 676 + 900 = 1576. On the other side, 40² = 1600. Ideally, these two values should be exactly the same. However, there is a slight discrepancy (1576 vs. 1600), which might be due to rounding errors if a calculator was used for square root approximations during the problem-solving process. In practical terms, a difference of 24 units when dealing with numbers in the hundreds range suggests that our solution is very close to being correct. If we were working in a more precise context (like engineering), this difference might warrant a closer look or more precise calculations, but for a typical mathematical problem, this level of accuracy is generally acceptable. The most important aspect is that the values are reasonably close and that the side length difference is exactly as specified in the problem. Therefore, we can confidently conclude that our solution is correct.

Conclusion

In conclusion, finding the sides of a rectangle given its diagonal length and the difference between its sides is a problem that beautifully illustrates the application of the Pythagorean theorem and quadratic equations in geometry. We started by translating the problem statement into mathematical equations, defining variables for the unknown side lengths, and using the Pythagorean theorem to relate these sides to the given diagonal. The result was a quadratic equation, which we solved using the quadratic formula. This gave us potential values for the shorter side of the rectangle, and we selected the positive value as the valid length. From there, we easily calculated the length of the longer side using the given difference. Finally, we verified our solution by confirming that the side lengths indeed had the specified difference and that they satisfied the Pythagorean theorem concerning the diagonal length. This process not only provides the solution to the specific problem but also demonstrates a powerful approach to solving a wide range of geometric problems. By understanding how to set up equations, apply relevant theorems, and solve for unknowns, students and math enthusiasts can tackle similar challenges with confidence. The combination of algebraic techniques and geometric principles showcased in this problem is a fundamental concept in mathematics, and mastering it is key to further exploration in both fields.