A polygon is such that when all diagonals from one vertex are drawn, eleven triangles are formed. How many sides does the polygon have?

Think about it. If the polygon has n vertices, how many lines can be drawn from one of them to each of the others?

To find the number of sides in the polygon, we need to determine the number of diagonals it has. Let's start by finding the formula for the number of diagonals in a polygon.

In any polygon with N sides, the number of diagonals can be calculated using the formula:
Number of diagonals = N(N-3)/2

Therefore, let's solve the equation:

N(N-3)/2 = 11

To solve the equation, we can multiply both sides by 2:

N(N-3) = 22

Expanding the equation, we get:

N² - 3N = 22

Rearranging the equation:

N² - 3N - 22 = 0

Now we can factorize the quadratic equation:

(N - 11)(N + 2) = 0

So we have two possible solutions for N:

N - 11 = 0 or N + 2 = 0

Solving the equations:

N = 11 or N = -2

Since the number of sides cannot be negative, we discard the solution N = -2.

Therefore, the polygon has 11 sides.