Question

If a polygon of \(n\) sides has \(560\) diagonals, then \(n=\)

• A. \(35\)
• B. \(36\)
• C. \(37\)
• D. \(38\)

Hint:

The number of diagonals in an \(n\)-sided polygon is always \(\frac{n(n-3)}{2}\), because each vertex joins with \(n-3\) non-adjacent vertices.

Answer 1

The Correct Option is A

Step 1: Recall the formula for diagonals of a polygon.
The number of diagonals in an \(n\)-sided polygon is given by
\[ \frac{n(n-3)}{2} \]
Given that the polygon has \(560\) diagonals, we get
\[ \frac{n(n-3)}{2}=560 \]

Step 2: Simplify the equation.
Multiplying both sides by \(2\),
\[ n(n-3)=1120 \]
\[ n^2-3n-1120=0 \]

Step 3: Solve the quadratic equation.
We factorize the quadratic equation:
\[ n^2-35n+32n-1120=0 \]
\[ n(n-35)+32(n-35)=0 \]
\[ (n-35)(n+32)=0 \]
Thus,
\[ n=35 \] or
\[ n=-32 \]
Since the number of sides of a polygon cannot be negative,
\[ n=35 \]

Step 4: Verification.
Substitute \(n=35\):
\[ \frac{35(35-3)}{2} \]
\[ =\frac{35\cdot32}{2} \]
\[ =35\cdot16 \]
\[ =560 \]
Hence, the value is correct.

Step 5: Final conclusion.
Therefore,
\[ \boxed{35} \]