Question 1151952
the number of diagonals of an n-sided polygon is: {{{n(n - 3) / 2}}}
the sum of the measure   of polygons angles is: {{{180(n-2)}}}

if the ratio of diagonals to the sum of the measure  of polygons angles is {{{1 : 26}}}, we have

{{{(n(n - 3) / 2):(180(n-2))=1:26}}}

{{{n(n - 3) / (2*180(n-2))=1:26}}}

{{{26n(n - 3) =360(n-2)}}}

{{{26n^2 - 78n=360n-720}}}

{{{26n^2 - 78n-360n+720=0}}}

{{{26n^2 - 438n+720=0}}}......factor

{{{2(13n^2 - 219n+360)=0}}}

{{{2(13n^2 - 24n-195n+360)=0}}}

{{{2((13n^2 -195n)- (24n -360))=0}}}

{{{2(13n(n -15)-24 (n -15))=0}}}

{{{2 (n - 15) (13n - 24) = 0}}}-> we need only whole number as solution , and it is

{{{(n - 15)= 0}}}=>{{{n=15}}}

 the number of sides of the polygon is {{{15}}}