Question

The ratio of number of sides of two regular polygons is 3:4 and the ratio of measures of their each interior angle is 8:9. What is the sum of the number of diagonals of both the polygons equal to?

​

Answer 1

Step-by-step explanation:

The ratio of number of sides of two regular polygons is 3:4 and the ratio of measures of their each interior angle is 8:9. What is the sum of the number of diagonals of both the polygons equal to?

Ratio of sides of two regular polygons = 3 : 4

Let sides of first polygon = 3n

and sides of second polygon = 4n

Sum of interior angles of first polygon

= (2 × 3n – 4) × 90° = (6n – 4) × 90°

And sum of interior angle of second polygon

= (2 × 4n – 4) × 90° = (8n – 4) × 90°

∴ ((6n – 4) × 90°)/((8n – 4) × 90°) = 2/3

⇒ (6n – 4)/(8n – 4) = 2/3

⇒ 18n – 12 = 16n – 8

⇒ 18n – 16n = -8 + 12

⇒ 2n = 4

⇒ n = 2

∴ No. of sides of first polygon

= 3n = 3 × 2 = 6

And no. of sides of second polygon

= 4n = 4n × 2 = 8

Answer 2

$\huge✎\fbox \orange{QUE} ST \fbox\green{ION}☟$

The ratio of number of sides of two regular polygons is 3:4 and the ratio of measures of their each interior angle is 8:9. What is the sum of the number of diagonals of both the polygons equal to?

$\huge✍︎\fbox\orange{ÂŇ}\fbox{SW} \fbox\green{ÊŘ}:$

Given⤵

The ratio of number of sides of two regular polygons is 3:4 and the ratio of measures of their each interior angle is 8:9. What is the sum of the number of diagonals of both the polygons equal to?

Let regular Polygon A have 3x sides while regular Polygon B has 4x sides.

Each exterior angle of

${A = \frac{360}{3x} or \frac{120}{x}}$

While That of

$B = \frac{360}{4x} or \: \frac{90}{x}$

Each interior angle of

${A = 180- \frac{120}{x} \: or \: \frac{180x - 120x}{x}}$

and that of

${B = 180- \frac{90}{x} \: or \: \frac{180x - 90}{x}.}$

The ratio of interior angles of A to that of B =

${ \frac{180x - 120}{x} : \frac{180x - 90}{x} = 8:9 \: \: \: Or}$

${9(180x-120) = 8(180x-90)}$

${1620x-1080 = 1440x-720, or}$

${1620x-1440x = 1080–720, or}$

${180x = 360, or}$

${x = 2}$

Polygon A has 6 sides and Polygon has 8 sides.

${Polygon \: A \: has \: \: 6\times\frac{6 - 3}{2} = 9 \: diagonals}$

$while \: B \: has \: {8 \times\frac{8 - 3}{2} = 20.}$

The sum of the diagonals of Polygons A and B is

${9+20 = 29.}$

$\footnotesize\bold\red{\overbrace{\underbrace \mathbb\blue{Please \: Mark \: As \: The \: Brainliest\: If\: It\: Satisfied\:You}}}$

꧁Hopes so that this will help you,꧂

♥️With Love♥️

✌︎Thanking you, your one of the Brother of your 130 Crore Indian Brothers and Sisters.✌︎

$\large\red{\underbrace{\overbrace\mathbb{\colorbox{aqua}{\colorbox{blue}{\colorbox{purple}{\colorbox{pink}{\colorbox{green}{\colorbox{gold}{\colorbox{silver}{\textit{\purple{\underline{\red{《❀PŘÎ᭄ŇCĒ✿》࿐}}}}}}}}}}}}}}$

$\large\red{\underbrace{\overbrace\mathbb{\colorbox{aqua}{\colorbox{blue}{\colorbox{purple}{\colorbox{pink}{\colorbox{green}{\colorbox{gold}{\colorbox{silver}{\textit{\purple{\underline{\red{《❀FÔLLÔW✿》࿐}}}}}}}}}}}}}}$