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How many DIAGONALS does a regular DECAGON (10-sided polygon) have?
A$10$
B$30$
C$35$
D$45$
Answer & Solution
Correct answer: C. $35$
1. A polygon with $n$ vertices has $\binom{n}{2}$ ways to connect two vertices — these are all the line segments.
2. Of these, $n$ are the SIDES of the polygon (consecutive vertex pairs).
3. The rest are DIAGONALS: $\binom{n}{2} - n$.
4. For a decagon ($n = 10$): diagonals = $\binom{10}{2} - 10 = 45 - 10 = 35$.
5. Useful formula: number of diagonals of an $n$-gon = $\dfrac{n(n-3)}{2}$. Check: $10 \cdot 7 / 2 = 35$ ✓.
6. Option A is just $n$ (the sides count). Option B is half by mistake. Option D is $\binom{10}{2}$ — forgot to subtract sides.
_Source: NCERT Class 11 Mathematics, Ch 6, §6.4 + Geometry applications, p. 12._
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