A polygon has 27 diagonals. How many sides does it have ? (a) 7 (b) 8 (c) 9 (d) 12

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Solution

$∴ \frac{n (n - 3)}{2} = 27 \Rightarrow n (n - 3) = 54 n^{2} - 3 n - 54 = 0 \Rightarrow n^{2} - 9 n + 6 n - 54 = 0 \Rightarrow n (n - 9) + 6 (n - 9) = 0 \Rightarrow (n - 9) (n + 6) = 0 E i t h e r n - 9 = 0, t h e n n = 9 o r n + 6 = 0, t h e n n = - 6 but it is not possible being negative ∴ No. of sides = 9$

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Tick (✓) the correct answer:
A polygon has $27$ diagonals. How many sides does it have?
(a) $7$
(b) $8$
(c) $9$
(d) $12$

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Each interior angle of a polygon is $135^{\circ}$ . How many sides does it have?

$(a) 8$

$(b) 7$

$(c) 6$

$(d) 10$

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