Solution: Let $ a + b = 10 $ and $ a^2 + b^2 = 58 $. Compute $ (a + b)^2 = 100 = a^2 + 2ab + b^2 = 58 + 2ab $. Solve for $ ab $: $ 2ab = 42 \Rightarrow ab = 21 $. The area is $ \frac{1}{2}ab = \boxed{\frac{21}{2}} $.

["Solving for Area Using Algebra: A Clear Path from Equations to Final Result", "When faced with mathematical problems involving two variables constrained by symmetric equations like $ a + b = 10 $ and $ a^2 + b^2 = 58 $, it can be tempting to rush ahead. However, a structured approach using algebraic identities not only ensures accuracy but also deepens understanding. This article walks through solving for the product $ ab $, a crucial step leading directly to computing the area defined by $ \frac{1}{2}ab $, demonstrated step by step.", "### Step-by-Step Breakdown", "We begin with the given equations:\n1. $ a + b = 10 $\n2. $ a^2 + b^2 = 58 $", "Our goal is to compute $ ab $, which then helps find the area $ \frac{1}{2}ab $.", "#### Step 1: Use the identity for the square of a sum\nRecall the algebraic identity:\n$$\n(a + b)^2 = a^2 + 2ab + b^2\n$$", "We already know $ a + b = 10 $, so:\n$$\n(a + b)^2 = 10^2 = 100\n$$", "Substitute into the identity:\n$$\n100 = a^2 + b^2 + 2ab\n$$", "#### Step 2: Plug in the known value\nWe are given $ a^2 + b^2 = 58 $, so:\n$$\n100 = 58 + 2ab\n$$", "#### Step 3: Solve for $ 2ab $\nSubtract 58 from both sides:\n$$\n100 - 58 = 2ab \Rightarrow 42 = 2ab\n$$", "Divide by 2:\n$$\nab = 21\n$$", "#### Step 4: Compute the area\nThe area in question is defined as:\n$$\n\ ext{Area} = \frac{1}{2}ab = \frac{1}{2} \ imes 21 = \boxed{\frac{21}{2}}\n$$", "### Why This Method Works\nBy leveraging the identity $ (a + b)^2 = a^2 + b^2 + 2ab $, we efficiently eliminate variables and isolate $ ab $ without solving complex equations. This technique is especially powerful in symmetric problems and forms the backbone of many algebraic and geometric proofs.", "### Conclusion\nUnderstanding how to manipulate equations like $ a + b $ and $ a^2 + b^2 $ using well-known identities is key to solving optimization, geometry, and algebra problems efficiently. From computing $ ab $ to determining the final area, each step builds logically on the previous—ensuring both correctness and clarity.", "So, when given $ a + b = 10 $ and $ a^2 + b^2 = 58 $, the area $ \frac{1}{2}ab $ simplifies neatly to $ \boxed{\frac{21}{2}} $."]