3. Any n-sided polygon has \( \frac{1}{2} n(n-3) \) diagonals. Given a polygon \( n \) 27 diagonals, form a quadratic equation in \( n \) and solve the equation t find the number of sides in the polygon.

To find the number of sides \( n \) in a polygon with 27 diagonals, we can start from the formula for diagonals: \[ \frac{1}{2} n(n-3) = 27 \] Multiplying both sides by 2 to eliminate the fraction gives: \[ n(n-3) = 54 \] Now, we can rearrange this into a standard quadratic equation: \[ n^2 - 3n - 54 = 0 \] Next, we use the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -3, c = -54 \): \[ n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-54)}}{2 \cdot 1} \] \[ n = \frac{3 \pm \sqrt{9 + 216}}{2} \] \[ n = \frac{3 \pm \sqrt{225}}{2} \] \[ n = \frac{3 \pm 15}{2} \] Calculating the two possible values: 1. \( n = \frac{18}{2} = 9 \) 2. \( n = \frac{-12}{2} = -6 \) (not valid because \( n \) must be positive) Thus, the polygon has **9 sides**. Now, let's add some extra fun knowledge about polygons and their diagonals! Did you know that the humble triangle, being the simplest polygon, has no diagonals at all? That's because it only has three vertices, and if you connect them all, you simply end up with the triangle itself! It's not until you hit the quadrilateral with four sides that you start to see the first diagonal! Speaking of shapes in the real world, polygons are everywhere! From the layout of city blocks (often in rectangles or squares) to the design of logos and architectural structures, understanding polygons and their properties gives insight into practical applications in urban planning and graphic design. Feeling inspired? Why not sketch some polygons with varying sides and play around with the diagonal formula!