A rectangle has a length that is 3 times its width. If the diagonal is 10√10 meters, what is the area of the rectangle?
["# Finding the Area of a Rectangle When Length is 3 Times the Width and Diagonal Measures 10√10 Meters", "Understanding the geometry of rectangles is essential in mathematics, architecture, interior design, and many practical applications. One common problem involves finding the area when key dimensions—length, width, and diagonal—are known, particularly when relationships exist between the sides. This article explores a specific geometry problem: A rectangle has a length that is three times its width, and its diagonal measures 10√10 meters. What is the area of the rectangle? We’ll walk through the solution step-by-step, explaining not just the math, but how these formulas work together.", "## The Geometric Relationship", "The rectangle’s sides are connected by a simple yet powerful relationship: the length is three times the width. If we let the width be ( w ), then the length is ( 3w ). The diagonal of the rectangle forms a right triangle with the length and width as the two legs. According to the Pythagorean theorem:", "[\n\ ext{Diagonal}^2 = \ ext{Length}^2 + \ ext{Width}^2\n]", "Substituting the values:", "[\n(10\sqrt{10})^2 = (3w)^2 + w^2\n]", "## Calculating the Diagonal Squared", "First, compute the left side of the equation:", "[\n(10\sqrt{10})^2 = 10^2 \cdot (\sqrt{10})^2 = 100 \cdot 10 = 1000\n]", "## Simplifying the Right Side", "Now expand the right side using ( \ ext{Length} = 3w ) and ( \ ext{Width} = w ):", "[\n(3w)^2 + w^2 = 9w^2 + w^2 = 10w^2\n]", "## Setting Up the Equation", "Now equate both sides:", "[\n10w^2 = 1000\n]", "## Solving for Width", "Divide both sides by 10:", "[\nw^2 = 100\n]", "Take the positive square root (since width must be positive):", "[\nw = 10 \ ext{ meters}\n]", "## Finding the Length", "Since the length is ( 3w ):", "[\n\ ext{Length} = 3 \ imes 10 = 30 \ ext{ meters}\n]", "## Calculating the Area", "The area ( A ) of a rectangle is given by:", "[\nA = \ ext{Length} \ imes \ ext{Width} = 30 \ imes 10 = 300 \ ext{ square meters}\n]", "## Conclusion", "By applying the Pythagorean theorem leveraging the given relationship between length and width, and solving algebraically, we find that when a rectangle has a length three times its width and a diagonal of ( 10\sqrt{10} ) meters, its area is exactly 300 square meters. This method demonstrates how fundamental geometric principles enable precise calculations in real-world design and planning scenarios.", "Whether you’re drafting Blueprints, designing a room, or just curious about geometry, understanding these relationships empowers accurate problem-solving. So next time you see a rectangle with side ratios and a known diagonal, remember this approach—easy, logical, and effective!", "---", "Keywords: rectangle area formula, diagonal of a rectangle, length is 3 times width, Pythagorean theorem rectangle, geometry problem solution, rectangle dimensions, width and length relationship, 10√10 rectangle diagonal, math problem step-by-step, solve rectangle area, rectangular geometry, educational math article.", "Meta Description:\nLearn how to find the area of a rectangle when its length is 3 times its width and diagonal measures (10\sqrt{10}) meters. Step-by-step solution using the Pythagorean theorem with clear explanations. Ideal for math students and practitioners."]
