Thus, the length of the hypotenuse is \(\boxed{15}\) and the area is \(\boxed{54}\).

["# Finding a Right Triangle with Hypotenuse 15 and Area 54: A Step-by-Step Solution", "When studying right triangles, a common challenge arises: given only the hypotenuse and the area, how do we determine the lengths of the other two sides? In this article, we’ll explore how to find the dimensions of a right triangle where the hypotenuse measures exactly (\boxed{15}) and the area is (\boxed{54}). Understanding this problem enhances geometric reasoning and problem-solving skills—skills highly valued in math and engineering fields.", "## Understanding the Problem", "We are given:\n- Hypotenuse (c = 15)\n- Area (A = 54)", "Let the two legs of the right triangle be (a) and (b). For a right triangle with legs (a) and (b) and hypotenuse (c), the following equations hold:", "1. Pythagorean Theorem:\n [\n a^2 + b^2 = c^2\n ]\n Substituting (c = 15):\n [\n a^2 + b^2 = 15^2 = 225\n ]", "2. Area of a Right Triangle:\n [\n A = \frac{1}{2}ab = 54\n ]\n Multiply both sides by 2:\n [\n ab = 108\n ]", "Now we have a system of two equations:\n[\n\begin{cases}\na^2 + b^2 = 225 \\nab = 108\n\end{cases}\n]", "## Using Algebra to Solve for (a) and (b)", "To find (a) and (b), we use the identity:\n[\n(a + b)^2 = a^2 + b^2 + 2ab\n]\nSubstitute known values:\n[\n(a + b)^2 = 225 + 2(108) = 225 + 216 = 441\n]\nTake the square root:\n[\na + b = \sqrt{441} = 21\n]", "Now we have:\n[\na + b = 21 \\nab = 108\n]", "These are the sum and product of the roots of a quadratic equation. Let (a) and (b) be the solutions to:\n[\nx^2 - (a + b)x + ab = 0 \Rightarrow x^2 - 21x + 108 = 0\n]", "Solve using the quadratic formula:\n[\nx = \frac{21 \pm \sqrt{(-21)^2 - 4 \cdot 1 \cdot 108}}{2}\n= \frac{21 \pm \sqrt{441 - 432}}{2}\n= \frac{21 \pm \sqrt{9}}{2}\n= \frac{21 \pm 3}{2}\n]", "So,\n[\nx = \frac{21 + 3}{2} = 12 \quad \ ext{or} \quad x = \frac{21 - 3}{2} = 9\n]", "Thus, the two leg lengths are (a = 9) and (b = 12).", "## Confirming the Hypotenuse", "Verify using the Pythagorean theorem:\n[\n9^2 + 12^2 = 81 + 144 = 225 = 15^2\n]\nThe Pythagorean identity holds, confirming the triangle’s validity.", "## Verifying the Area", "Compute the area:\n[\n\frac{1}{2} \cdot 9 \cdot 12 = \frac{1}{2} \cdot 108 = 54\n]\nThe area is confirmed.", "## Why This Problem Matters", "Right triangles with specified hypotenuses and areas are foundational in trigonometry, engineering, architecture, and physics. Solving such problems strengthens algebraic manipulation and geometric intuition. This example illustrates how two constraints uniquely determine a right triangle—offering a clear model for similar challenges.", "## Conclusion", "For a right triangle with hypotenuse (\boxed{15}) and area (\boxed{54}), the legs measure (\boxed{9}) units and (\boxed{12}) units. This solution combines algebraic identities, the Pythagorean theorem, and verification steps to accurately determine the triangle’s missing sides while ensuring geometric consistency.", "---", "### TL;DR:\nA right triangle with hypotenuse 15 and area 54 has legs of 9 and 12. Using the Pythagorean theorem and area formula, we derived these values through systematic equations and algebraic solving—proving that precise geometric constraints yield unique solutions."]