The ratio of number two sides of two regular polygons is 3 : 4 and the ratio of the sum of their interior angles is 1 : 2. Find the number of sides of each polygon.
Answers
Answer:
The number of sides of the Polygons are 3 and 4.
Step-by-step explanation:
The ratio of number of sides of two regular polygons is 3:4
Let's assume that the proportionality constant is n
∴ The number of sides of the polygons are 3n and 4n.
Now, from one vertex of n sided polygon, we can draw (n-3) diagonals. This is because the two adjacent vertices form adjacent sides and not diagonals. So, from the n number of vertices, one is the vertex itself from where we are drawing the diagonals, and we have just discussed about the two adjacent vertices.
With n - 3 diagonals from one vertex, we will get (n - 2) triangles dividing all the interior angles of the n sided polygon amongst themselves.
Therefore, the sum of the interior angles of n sided regular polygon is (n - 2)x180°
Therefore, the sum of the interior angles of 3n sided regular polygon is (3n - 2)x180°
And, the sum of the interior angles of 4n sided regular polygon is (4n - 2)x180°
According to the problem:
∴
∴ and
∴ the number of sides of the Polygons are 3 and 4.
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