Let the sides of the rectangle be \( a = 5 \) inches and \( b \) inches. The diagonal \( d = 13 \) inches. By the Pythagorean theorem, we have:

["Understanding Rectangles: Using the Pythagorean Theorem with Dimensions ( a = 5 ) inch and ( d = 13 ) inch", "In geometry, rectangles are fundamental shapes defined by four right angles and opposite sides that are equal in length. One key property of a rectangle is that its diagonal divides it into two right triangles. When given two sides and the diagonal, the Pythagorean theorem provides a powerful tool to verify or solve for unknown dimensions.", "Let’s explore a classic example:\nLet the width of the rectangle be ( a = 5 ) inches, the length be ( b ) inches, and the diagonal ( d = 13 ) inches.", "### Applying the Pythagorean Theorem", "In a rectangle, the diagonal forms the hypotenuse of a right triangle with sides ( a ) and ( b ). According to the Pythagorean theorem:\n[\na^2 + b^2 = d^2\n]", "Substitute the known values:\n[\n5^2 + b^2 = 13^2\n]", "Calculate the squares:\n[\n25 + b^2 = 169\n]", "Solve for ( b^2 ) by subtracting 25 from both sides:\n[\nb^2 = 169 - 25 = 144\n]", "Now take the square root:\n[\nb = \sqrt{144} = 12 \ ext{ inches}\n]", "### Conclusion", "Thus, the rectangle has side lengths ( a = 5 ) inches and ( b = 12 ) inches, with a diagonal of 13 inches, confirming the Pythagorean relationship:\n[\n5^2 + 12^2 = 13^2\n]", "This relationship not only verifies geometric accuracy but also illustrates how mathematical principles underpin real-world measurements and design. Whether in construction, carpentry, or drafting, using the Pythagorean theorem ensures precision when working with rectangular shapes.", "---", "Key takeaways:\n- For a rectangle with sides ( a ) and ( b ), and diagonal ( d ), apply ( a^2 + b^2 = d^2 )\n- Solving for unknowns allows confident measurement and design\n- The 5–12–13 triangle is a well-known Pythagorean triple, ideal for quick verification", "Use this method whenever determining dimensions or checking structural integrity involving rectangular forms.", "Keywords: rectangle Pythagorean theorem, diagonal of rectangle calculation, solve for unknown rectangle side, a = 5 inches, b = unknown, diagonal d = 13 inches, geometry problem, right triangle in rectangles."]