If the ratio of the number of sides of two regular polygons is 5:8 and the ratio of the sum of their interior angles is 13:22, then the ratio of their number of diagonals is
Answers
Step-by-step explanation:
Given :-
The ratio of the number of sides of two regular polygons is 5:8 and the ratio of the sum of their interior angles is 13:22.
To find:-
Find the ratio of their number of diagonals?
Solution :-
Given that
The ratio of the number of sides of two regular polygons = 5:8
Let they be 5X and 8X
The number of sides in the first polygon = 5X
The number of sides in the second polygon = 8X
We know that
The sum of the interior angles in a regular polygon of n sides is (n-2)×180°
The sum of the interior angles in the first polygon of 5X sides = (5X-2)×180°
The sum of the interior angles in the second polygon of 8X sides = (8X-2)×180°
The ratio of the sum of the interior angles of the two polygons = [(5X-2)×180°]:[(8X-2)×180°]
=> (5X-2) : (8X-2)
According to the given problem
The ratio of the sum of their interior angles = 13:22
=> (5X-2) : (8X-2) = 13:22
=> (5X-2) / (8X-2) = 13 / 22
On applying cross multiplication then
=>22(5X-2) = 13(8X-2)
=> 110X - 44 = 104X - 26
=> 110X - 104X = -26 + 44
=> 6X = 18
=>X = 18/6
=> X = 3
So,
The number of sides in the first polygon
= 5(3) = 15
The number of sides in the second polygon
= 8(3) = 24
We know that
The number of diagonals in a regular polygon of n sides = n(n-3)/2
Number of diagonals in the first polygon
= 15(15-3)/2
= 15(12)/2
= 15(6)
= 90
Number of diagonals in the first polygon
= 24(24-3)/2
= 24(21)/2
= 12(21)
= 252
The ratio of the diagonals of the two polygons
= 90 : 252
= 90 / 252
= (5×18) / (14×18)
= 5/14
= 5:14
Answer:-
The ratio of the diagonals of the given polygons is 5 : 14
Used formulae:-
The number of sides in a polygon is 'n'
- The sum of the interior angles in the polygon is (n-2)×180°
- The number of diagonals = n(n-3)/2
- a:b can be written as a/b
Given :- If the ratio of the number of sides of two regular polygons is 5:8 and the ratio of the sum of their interior angles is 13:22, then the ratio of their number of diagonals is ?
Solution :-
Let us assume that, sides of given two regular polygons are 5x and 8x respectively .
we know that,
- sum of all interior angles of a regular polygon = (n - 2) * 180° .
A/q,
→ (5x - 2) * 180° / (8x - 2) * 180° = 13/22
→ 22(5x - 2) = 13(8x - 2)
→ 110x - 44 = 104x - 26
→ 110x - 104x = 44 - 26
→ 6x = 18
→ x = 3 .
then,
→ Sides of first regular polygon = 5x = 5 * 3 = 15
→ Sides of second regular polygon = 8x = 8 * 3 = 24
now, we know that,
- Total number of diagonals in a regular polygon with n sides is = n(n - 3)/2
therefore,
→ Required ratio = 15(15 - 3)/2 : 24(24 - 3)/2
→ Required ratio = 15 * 12 = 24 * 21
→ Required ratio = 15 : 42
→ Required ratio = 5 : 14 (Ans.)
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