A rectangle's diagonal is 10 cm, and one side is 6 cm. What is the area of the rectangle?
["Finding the Area of a Rectangle When Its Diagonal and One Side Are Known", "Understanding the relationship between a rectangle’s sides and its diagonal is essential in geometry. For anyone curious about how dimensions affect area, consider a rectangle where the diagonal measures 10 cm and one side is 6 cm. This scenario provides a perfect opportunity to apply the Pythagorean theorem and calculate the area.", "### How the Diagonal Relates to Rectangle Sides", "A rectangle’s diagonal forms two right triangles with its adjacent sides. Using the Pythagorean theorem:", "[\n\ ext{Diagonal}^2 = \ ext{Length}^2 + \ ext{Width}^2\n]", "Given:\n- Diagonal = 10 cm\n- One side (say, length) = 6 cm\nLet the unknown side (width) be ( b ).", "Substitute into the formula:", "[\n10^2 = 6^2 + b^2\n]", "[\n100 = 36 + b^2\n]", "[\nb^2 = 100 - 36 = 64\n]", "[\nb = \sqrt{64} = 8 \ ext{ cm}\n]", "### Calculating the Area of the Rectangle", "The area of a rectangle is given by:", "[\n\ ext{Area} = \ ext{Length} \ imes \ ext{Width}\n]", "Substituting the known values:", "[\n\ ext{Area} = 6,\ ext{cm} \ imes 8,\ ext{cm} = 48,\ ext{cm}^2\n]", "### Conclusion", "When a rectangle’s diagonal is 10 cm and one side measures 6 cm, the other side is 8 cm, using the Pythagorean theorem. The area of the rectangle is 48 cm². This simple yet powerful relationship underscores the importance of geometric principles in solving real-world measurement problems.", "Whether you’re a student, teacher, or DIY enthusiast, knowing how to derive side lengths and area from diagonal and one side is invaluable in fields like architecture, design, and engineering."]
