solve it---**Question Stem:** The figure shows a regular hexagon. If AB = 12, find the area of triangle AEC. Express the answer in surd form. **Geometric Diagram Description:** Type: Regular hexagon with vertices labeled A, B, C, D, E, F arranged in counterclockwise order. Lines: - Sides of the hexagon: AB, BC, CD, DE, EF, FA. - Diagonals drawn: AC and AE. - Triangle formed: Triangle AEC, with vertices A, E, and C, and sides AE, EC, and AC. - The vertices A, B, C, D, E, F are positioned visually such that AB is the top side, BC is the top-right, CD is the bottom-right, DE is the bottom, EF is the bottom-left, and FA is the top-left side. - Lines AC and AE connect vertex A to vertices C and E respectively. **Given Information:** - The figure is a regular hexagon. - Side length AB = 12. **Question Asked:** Find the area of triangle AEC. **Output Format Requirement:** Express the answer in surd form. **Other Information:** None. (Ignoring handwritten annotations as they are not part of the problem statement or formal diagram labels).

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We have a regular hexagon with vertices labeled A, B, C, D, E, F in counterclockwise order. The side length AB equals 12. We need to find the area of triangle AEC, which is formed by connecting vertices A, E, and C. Let me highlight this triangle and the given side length. Now let's find the lengths of the sides of triangle AEC. In a regular hexagon, there are two types of diagonals. The long diagonal, like AE, equals twice the side length, so AE equals 24. The short diagonals, like AC and EC, equal the side length times square root of 3, so both AC and EC equal 12 square root of 3. This makes triangle AEC an isosceles triangle. Since triangle AEC is isosceles with AC equals EC, we can find its area by calculating the height. Let M be the midpoint of base AE, so AM equals ME equals 12. The line CM is the height of the triangle. In the right triangle AMC, we can use the Pythagorean theorem. AC squared equals AM squared plus CM squared. Substituting our values: 12 square root 3 squared equals 12 squared plus CM squared. This gives us 432 equals 144 plus CM squared, so CM squared equals 288, and CM equals 12 square root 2. Now we can calculate the area of triangle AEC using the formula: Area equals one half times base times height. We have the base AE equals 24 and the height CM equals 12 square root 2. So the area equals one half times 24 times 12 square root 2, which simplifies to 12 times 12 square root 2, giving us 144 square root 2. This is our final answer expressed in surd form. Let's summarize our solution. We were asked to find the area of triangle AEC in a regular hexagon where AB equals 12. First, we found the side lengths: AE equals 24 as a long diagonal, and AC and EC both equal 12 square root 3 as short diagonals. This made triangle AEC isosceles. We then calculated the height from C to base AE, which equals 12 square root 2. Finally, using the area formula, we got our answer: 144 square root 2, expressed in surd form as required.