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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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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Verified answer
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
Given :-
To find :-
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
[tex]\\[/tex]
and Let, the sum of their interior angles be
2 y and 3 y
[tex]\\[/tex]
then,
Using formula for sum of interior angles of a polygon
[tex]\\[/tex]
In first polygon
[tex]\\[/tex]
[tex]\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)}\\\\[/tex]
Also,
In second polygon
[tex]\\[/tex]
[tex]\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}\\\\[/tex]
so,
Number of sides of polygons will be
[tex]\\[/tex]
→ 3 x = 3 ( 2 ) = 6
and
→ 4 x = 4 ( 2 ) = 8
[tex]\\[/tex]
Hence,
▶ Number of sides of polygons are 6 and 8 .