Question

A convex polygon of n sides has 135 diagonals in total, find the value of n.​

Answer 1

Step-by-step explanation:

Solution :– n= -15;n=18 ==>Thus, we don't have any negative side.So, the number of sides are 18.

hope it helps u

Answer 2

$\mathsf \red{Given \: }$

$\mathsf{Diagonals \: of \: a \: polygon \: = \: 135}$

$\\ \mathsf{formula \: to \: find \: sides,\: if \: diagonals \: are \: given} \\ \\ \boxed{ \mathsf \blue{ \frac{n(n - 3)}{2} = no. \: of \: diagonals}}$

$\mathsf \red{then,}$

$\mathsf{ \frac{n(n - 3)}{2} = 135}$

$\mathsf{n(n - 3) = 135 \times 2} \\ \\ \mathsf{{n}^{2} - 3n = 270 \: \: \: \: \: \: \: \: \: } \\ \\ \mathsf{ {n}^{2} - 3n - 270 = 0}$

$\mathsf \red{by \: using \: quadratic \: formula : } \\ \\ \boxed{ \large\mathsf \blue{x = \: \frac{\: \: - b \: \pm \: \sqrt{ {b}^{2} - 4ac } }{2a} }}$

$\mathsf{a = 1, \: b = - 3 \:, \: c = - 270 } \\ \\ \mathsf \red{by \: substituing \: we \: get : \: \: \: \: \: \: \: \: \: }$

$\mathsf{n = \frac{ - ( - 3) \: \pm \: \sqrt{ {( - 3)}^{2} - 4(1)( - 270) } }{2(1)} } \\ \\ \mathsf{n = \frac{3 \: \pm \: \sqrt{9 + 1080} }{2} } \\ \\ \mathsf{n = \frac{3 \: \pm \: \sqrt{1089} }{2} } \\ \\ \mathsf{n = \frac{3 \: \: \pm \: 33}{2} } \\ \\ \mathsf{n = \frac{3 + 33}{2} \: \: \: ; \: \: \: n = \frac{3 - 33}{2} } \\ \\ \mathsf{n = \frac{36}{2} \: \: ; \: \: n = \frac{ - 30}{2} } \\ \\ \mathsf{n \: = \: 18 \: \: \: ; \: n = - 15 } \\ \\ \mathsf \red{sides \: of \: a \: polygon \: will \: never \: be \: as \: negative \: ; \: so} \\ \\ \boxed{\mathsf{n = 18}}$

$\mathsf\red{\therefore \: sides \: of \: a \: Convex \: polygon \: = \: 18}$