Question

A polygon has 170 diagonals. The number of sides of the polygon, is

• A. 20
• B. 24
• C. 25
• D. 30
• E. 28

Hint:

Math Tip: To quickly solve $n(n-3) = 340$ without the full quadratic formula, look for two numbers close to $\sqrt{340}$ (which is around $18.4$) that differ by exactly 3. Trying 17 and 20 immediately yields the result.

Answer 1

The Correct Option is A

Concept:
Geometry / Combinatorics - Diagonals of a Polygon.
The number of diagonals in an $n$-sided polygon is given by the formula $\frac{n(n-3)}{2}$.
Step 1: Set up the equation using the given formula.
Let $n$ be the number of sides of the polygon. $$ \frac{n(n-3)}{2} = 170 $$
Step 2: Simplify to form a quadratic equation.
Multiply both sides by 2: $$ n(n-3) = 340 $$ $$ n^2 - 3n - 340 = 0 $$
Step 3: Factorize the quadratic equation.
We need two numbers that multiply to $-340$ and add to $-3$.

• The numbers are $-20$ and $17$.

$$ (n - 20)(n + 17) = 0 $$
Step 4: Solve for n.
This gives two possible values for $n$: $$ n = 20 \quad \text{or} \quad n = -17 $$
Step 5: Determine the valid number of sides.
Since a polygon cannot have a negative number of sides, discard $n = -17$. $$ n = 20 $$