Use the Pythagorean theorem to compute the diagonal \(d\) of the rectangle:
["# Using the Pythagorean Theorem to Compute the Diagonal of a Rectangle", "When working with rectangles in geometry, one essential measurement is the length of the diagonal. Calculating this diagonal accurately can simplify problems in construction, design, and navigation. A powerful and practical tool for this task is the Pythagorean theorem, a foundational principle in Euclidean geometry. In this article, we explore how to use the Pythagorean theorem to determine the diagonal (d) of a rectangle with given side lengths.", "## What Is the Pythagorean Theorem?", "The Pythagorean theorem 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. Mathematically, this is expressed as:", "[\nd^2 = a^2 + b^2\n]", "where:\n- (d) is the length of the diagonal (hypotenuse),\n- (a) and (b) are the lengths of the rectangle’s adjacent sides.", "## Step-by-Step: Computing the Diagonal Using the Pythagorean Theorem", "Consider a rectangle with sides (a) and (b). The diagonals of a rectangle are equal in length and bisect each other at 90°, forming two right triangles with the sides and the diagonal as sides.", "### Step 1: Identify Rectangle Side Lengths\nLet the lengths of the sides be (a) and (b). These are the two perpendicular dimensions of the rectangle.", "### Step 2: Apply the Pythagorean Theorem\nSince the diagonal forms the hypotenuse of a right triangle with sides (a) and (b), plug these values into the theorem:", "[\nd^2 = a^2 + b^2\n]", "### Step 3: Solve for (d)\nTo get the diagonal length, take the square root of both sides:", "[\nd = \sqrt{a^2 + b^2}\n]", "### Step 4: Example Calculation\nSuppose we have a rectangle with side lengths (a = 3) meters and (b = 4) meters. Applying the formula:", "[\nd = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \ ext{ meters}\n]", "Thus, the diagonal measures 5 meters. This classic 3-4-5 triangle example confirms the theorem’s accuracy in real-world applications.", "## Why This Method Matters", "Using the Pythagorean theorem to compute the rectangle diagonal ensures precision without complex measurements. It applies universally whenever dealing with right triangles formed by rectangle edges and diagonals. This concept is valuable both in academic studies and practical applications like architecture, carpentry, and computer graphics, where accurate spatial calculations are indispensable.", "## Summary", "To compute the diagonal (d) of a rectangle using the Pythagorean theorem:\n1. Identify the two adjacent side lengths (a) and (b).\n2. Apply (d = \sqrt{a^2 + b^2}).\n3. Calculate to obtain the exact diagonal length.", "Mastering this approach equips you with a straightforward yet powerful method for solving geometry problems efficiently. Whether you’re designing a space or analyzing structural dimensions, the Pythagorean theorem remains a timeless tool for accurate measurement.", "---", "Keywords: Pythagorean theorem, diagonal of a rectangle, compute diagonal, geometry formula, rectangle sides, right triangle application, geometry tutorial, forex diagonal calculation, math in rectangles, diagonal length formula."]
