Question

The number of diagonals of a polygon is 35. If A, B are two distinct vertices of this polygon, then the number of all those triangles formed by joining three vertices of the polygon having AB as one of its sides is:

• A. 1
• B. 8
• C. 10
• D. 12

Answer 1

The Correct Option is C

Step 1: Understanding the Problem:
We are given that the number of diagonals of a polygon is 35. The number of diagonals of a polygon with \(n\) sides is given by the formula:

$ D = \frac{n(n-3)}{2} $

Step 2: Solving for \(n\):
We are told that the number of diagonals is 35, so we can substitute this into the formula:

$ \frac{n(n-3)}{2} = 35 $
Multiplying both sides by 2:

$ n(n-3) = 70 $

Step 3: Simplifying the Equation:
Expanding the left-hand side:

$ n^2 - 3n = 70 $
Rearranging the equation:

$ n^2 - 3n - 70 = 0 $

Step 4: Solving the Quadratic Equation:
We will solve the quadratic equation using the quadratic formula:

$ n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-70)}}{2(1)} $

First, simplify inside the square root:

$ n = \frac{3 \pm \sqrt{9 + 280}}{2} = \frac{3 \pm \sqrt{289}}{2} = \frac{3 \pm 17}{2} $

Step 5: Finding \(n\):
Taking the two possible solutions:

$ n = \frac{3 + 17}{2} = 10 $
or
$ n = \frac{3 - 17}{2} = -7 $

Step 6: Conclusion:
The number of sides of the polygon is \(n = 10\), as the number of sides cannot be negative.

Step 7: Finding the Number of Triangles:
To find the number of triangles that can be formed by selecting three vertices of the polygon, we use the combination formula:

Number of triangles = \( \binom{n}{3} = \frac{n(n-1)(n-2)}{6} \)
Substituting \(n = 10\):

$ \binom{10}{3} = \frac{10 \times 9 \times 8}{6} = 120 $

Step 8: Using the Given Condition:
We are given that \(A\) and \(B\) are two distinct vertices of the polygon, and we need to find the number of triangles that include \(AB\) as one of their sides. To form such a triangle, we need to choose one more vertex from the remaining 8 vertices.
Thus, the number of such triangles is 8.

Final Answer:
The number of triangles formed by joining three vertices of the polygon with \(AB\) as one of its sides is 8.