Question

The number of sides of two regular polygons are in the ratio 3:4.The sums of the interior angles of the two polygons are in the ratio 2:3.Calculate the number of sides of the two polygons.

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Answer 1

Sum of interior angles regular polygons = (n−2)∗180

sides of two regular polygons= x and y

So, x/y=3/4

x=3y/4

Sum of interior angles regular polygons (x)=(x−2)∗180

Sum of interior angles regular polygons (y)=(y−2)∗180

(x−2)∗180/(y−2)∗180=2/3

x−2/y−2=2/3

3y/4−2/y−2=2/3

3y−8/y−2=8/3

9y−24=8y−16

y=8

x=3y/4=3∗8/4=6

Answer (A) 6 and 8

Answer 2

Given :-

• Number of sides of two regular polygons are in the ratio 3 : 4

• Sums of interior angles of two polygons are in the ratio 2 : 3

To find :-

• Number of sides of the two polygons

Knowledge required :-

The sum of the interior angles of a polygon with n sides is given by

→  ( n - 2 ) × 180°

Solution :-

Let,

no. of sides in Two polygons be

3 x  and 4 x

and Let, the sum of their interior angles be

2 y  and 3 y

then,

Using formula for sum of interior angles of a polygon

In first polygon

$\implies\sf{(3x-2) 180^{\circ}=2y}\\\\\\\implies\sf{540 x - 360 = 2y}\\\\\\ \sf{dividing\:by\:2\:both\:sides} \\\\\\\implies\sf{270x-180 = y \;\;\;......\; eqn(1)}$

Also,

In second polygon

$\implies\sf{(4x-2)180^{\circ}=3y}\\\\\\\implies\sf{720x - 360 = 3 y}\\\\\\\sf{dividing\:by\:3\:both\:sides}\\\\\\\implies\sf{240x-120=y}\\\\\\\sf{using\:eqn(1)}\\\\\\\implies\sf{240x - 120 =270x-180 }\\\\\\\implies\sf{240x-270x=-180+120}\\\\\\\implies\sf{-30x = -60}\\\\\\\implies\sf{30x=60}\\\\\\\implies\boxed{\boxed{\large{\red{\sf{x=2}}}}}\red{\bigstar}$

so,

Number of sides of polygons will be

→  3 x = 3 ( 2 ) = 6

and

→ 4 x = 4 ( 2 ) = 8

Hence,

▶ Number of sides of polygons are 6 and 8 .