After increasing each leg by 2 cm, the new legs are 10 cm and 17 cm. The new area is:

["Title: Use Geometry to Calculate New Area After Increasing Leg Lengths – Step-by-Step Explanation", "When working with geometric shapes involving rectangular or trapezoidal bases—common in architecture, engineering, or design—alterations to key dimensions like leg (side) lengths directly impact area calculations. Let’s explore a practical example: increasing each leg of a geometric figure by 2 cm, resulting in new leg measurements of 10 cm and 17 cm. We’ll determine the new area and explain the process clearly for better understanding.", "---", "### Problem Setup", "Initially, one leg measurement is unknown, and after increasing each leg by 2 cm, the new lengths become:", "- New leg 1 = 10 cm\n- New leg 2 = 17 cm", "Since each leg increased by 2 cm, the original leg lengths were:", "- Original leg 1 = 10 cm – 2 cm = 8 cm\n- Original leg 2 = 17 cm – 2 cm = 15 cm", "Assuming these legs form dimensions of a geometric figure—such as a trapezoid or rectangle—let’s explore how area changes after the adjustment.", "---", "### Understanding the Geometry", "For a simple rectangular or trapezoidal shape, legs often correspond to parallel or perpendicular sides critical to calculating area.", "In a trapezoid, the area formula is:", "[\n\ ext{Area} = \frac{(a + b)}{2} \ imes h\n]", "Where ( a ) and ( b ) are the lengths of the parallel sides (legs in your case), and ( h ) is the height. However, if it’s a rectangle with both legs perpendicular and equal in certain configurations (or if the height relates straightforwardly), area may depend directly on leg lengths.", "But here, the problem specifies each leg increased by 2 cm, implying a direct update based on geometric context—likely a trapezoid or similar structure where leg length influences base or height proportionally.", "However, based on the given data (original legs: 8 cm and 15 cm → new legs: 10 cm and 17 cm), a key insight emerges:", "- The increase of 2 cm implies a linear scaling or adjustment along a consistent dimension.", "---", "### Calculating the New Area", "Since both legs increased consistently by 2 cm, and area depends directly on leg lengths when shape geometry preserves proportionality (e.g., increasing parallel sides in a trapezoid while keeping height constant), we can model the area increase accordingly.", "Assuming original legs were bases ( a = 8 ) cm and ( b = 15 ) cm, and new legs are:", "- ( a_{\ ext{new}} = 10 ) cm\n- ( b_{\ ext{new}} = 17 ) cm", "Compute original area (using trapezoid formula with assumed height ( h )):", "[\n\ ext{Original Area} = \frac{(8 + 15)}{2} \ imes h = \frac{23}{2} \ imes h = 11.5h , \ ext{cm}^2\n]", "New area:", "[\n\ ext{New Area} = \frac{(10 + 17)}{2} \ imes h = \frac{27}{2} \ imes h = 13.5h , \ ext{cm}^2\n]", "Now, compute the change:", "[\n\ ext{Increase in Area} = 13.5h - 11.5h = 2.0h , \ ext{cm}^2\n]", "Thus, the new area exceeds the original by 2h cm², directly tied to the 2 cm leg increase and average base.", "---", "### Practical Example with Assumed Height", "Suppose the height ( h = 10 ) cm (for illustration):", "- Original area: ( 11.5 \ imes 10 = 115 ) cm²\n- New area: ( 13.5 \ imes 10 = 135 ) cm²\n- Difference: ( 135 - 115 = 20 ) cm², which matches ( 2.0 \ imes 10 = 20 ) cm²", "So, with each leg increased by 2 cm and a constant height, the area increases by 2 cm × height.", "---", "### Real-World Applications", "This principle applies in:", "- Engineering design, where beam or support leg dimensions affect cross-sectional area and structural strength\n- Architecture, updating floor or wall dimensions in scaled renovations\n- 3D modeling, adjusting vertices on digitized shapes for accurate volume estimation\n- Gardening or construction, recalculating plots or fencing based on adjusted boundary dimensions", "---", "### Summary", "- Original leg lengths: 8 cm and 15 cm\n- Each leg now: 10 cm and 17 cm (+2 cm)\n- Using trapezoid area formula: ( \frac{(a + b)}{2} \ imes h )\n- Area increase depends on average base and assumed height: ( 2h ) cm²\n- Example: height of 10 cm → new area 135 cm², up by 20 cm²", "---", "### Final Answer", "The new area after increasing both legs by 2 cm—from 10 cm and 17 cm—is calculated using the updated average of the leg bases, resulting in an increase of 2 times the original height. If the height is 10 cm, the new area is 135 cm², demonstrating how geometric adjustments directly influence measurable properties like area.", "Understanding these relationships helps in precise design, cost estimation, and spatial analysis across multiple fields.", "---", "Keywords: trapezoid area calculation, leg length increase, geometric area update, height influence on area, dimensional change impact, architectural geometry, practical geometry, area change formula", "Meta Description: Learn how increasing each leg by 2 cm (from 8 cm and 15 cm to 10 cm and 17 cm) affects area using trapezoid area formula—perfect for architects and engineers. Calculate new area now!"]