A rectangle has a length of 12 cm and a width of 5 cm. A right triangle is formed by drawing a diagonal across the rectangle. Find the length of the diagonal.
["Finding the Diagonal of a Rectangle: A Case Study with Dimensions 12 cm by 5 cm", "When studying geometry, one of the most fundamental problems involves calculating the diagonal of a rectangle. This scenario is perfectly illustrated with a rectangle that has a length of 12 cm and a width of 5 cm. Understanding how to compute the diagonal not only reinforces the application of the Pythagorean theorem but also enhances spatial reasoning skills essential in mathematics and engineering.", "---", "### Understanding the Problem", "A rectangle features opposite sides of equal length and four right angles. Drawing a diagonal from one corner to the opposite corner splits the rectangle into two congruent right triangles. Each triangle’s hypotenuse is the diagonal of the rectangle. In our example:", "- Length (longer side) = 12 cm\n- Width (shorter side) = 5 cm", "By applying the Pythagorean theorem — a cornerstone theorem in geometry — we can determine the length of the diagonal.", "---", "### The Pythagorean Theorem Explained", "The Pythagorean theorem states that in a right 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:", "[\nc^2 = a^2 + b^2\n]", "Where:\n- (c) is the length of the hypotenuse,\n- (a) and (b) are the lengths of the other two sides.", "This formula directly applies to the right triangle formed inside our rectangle.", "---", "### Applying the Formula to Our Rectangle", "Let:\n- (a = 12) cm (length),\n- (b = 5) cm (width),\n- (c = \ ext{diagonal} = ?)", "Substitute into the Pythagorean formula:", "[\nc^2 = 12^2 + 5^2\n]", "[\nc^2 = 144 + 25\n]", "[\nc^2 = 169\n]", "Now take the square root of both sides to solve for (c):", "[\nc = \sqrt{169} = 13 \ ext{ cm}\n]", "---", "### Why This Matters", "Knowing how to find the diagonal of a rectangle is crucial in various real-world applications — from designing furniture and architecture to engineering and computer graphics. Precisely calculating the diagonal ensures accuracy in measurements, fittings, and geometric modeling.", "---", "### Summary", "- A rectangle with a length of 12 cm and width of 5 cm forms two right triangles when a diagonal is drawn.\n- Using the Pythagorean theorem, the diagonal length comes out to 13 cm.\n- This simple yet powerful calculation is a foundational skill in geometry and applied mathematics.", "Whether you're a student, educator, or science enthusiast, mastering this concept helps unlock deeper geometric insights and practical problem-solving abilities.", "---", "Keywords: rectangle diagonal, Pythagorean theorem, triangle formed by diagonal, rectangle dimensions 12 cm 5 cm, calculate rectangle diagonal, right triangle geometry, geometry problem solution", "Meta Description: Learn how to find the diagonal of a rectangle with length 12 cm and width 5 cm using the Pythagorean theorem. Step-by-step solution and explanation."]
