Plane Geometry
- Ratio of the number of sides of two regular polygons is 5 : 6 and the ratio of their each interior angle is 24 : 25. Then the number of sides of these two polygons are
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Given , Ratio of the number of sides of two regular polygons = 5 : 6
and the ratio of their each interior angle = 24 : 25
Let the number of sides be 5y and 6y respectively.Then, (2 × 5y - 4) ÷ 5y = 24 (2 × 6y - 4) ÷ 6y 25 10y - 4 5y = 24 12y - 4 25 6y 
we know that , Each interior angle = 
(2n - 4)90° 
n ⇒ 5y - 2 × 6 = 24 5 6y - 2 25
Correct Option: C
Given , Ratio of the number of sides of two regular polygons = 5 : 6
and the ratio of their each interior angle = 24 : 25
Let the number of sides be 5y and 6y respectively.Then, (2 × 5y - 4) ÷ 5y = 24 (2 × 6y - 4) ÷ 6y 25 10y - 4 5y = 24 12y - 4 25 6y we know that , Each interior angle = (2n - 4)90° n ⇒ 5y - 2 × 6 = 24 5 6y - 2 25 ⇒ 5y - 2 = 4 6y - 2 5
⇒ 25y – 10 = 24y – 8
⇒ y = 10 – 8 = 2
∴ Number of sides = 10 and 12.
- Measure of each interior angle of a regular polygon can never be :
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According to question ,
Check through options ,Each interior angle = 2n - 4 × 90° n
If measure of each angle = 105°then, (2n - 4) × 90° = 105° n
Correct Option: B
According to question ,
Check through options ,Each interior angle = 2n - 4 × 90° n
If measure of each angle = 105°then, (2n - 4) × 90° = 105° n
⇒ (2n – 4) × 6 = 7n
⇒ 12n – 24 = 7n
⇒ 5n = 24n = 24 which is impossible. 5
- The sum of all interior angles of a regular polygon is twice the sum of all its exterior angles. The number of sides of the polygon is
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As we know that , Sum of interior angles = (2n – 4) × 90°
Sum of exterior angles = 360°
As per the given question ,
∴ (2n – 4) × 90° = 360° × 2
⇒ 2n – 4 = 2 × 360° ÷ 90 = 8
⇒ 2n – 4 = 8Correct Option: D
As we know that , Sum of interior angles = (2n – 4) × 90°
Sum of exterior angles = 360°
As per the given question ,
∴ (2n – 4) × 90° = 360° × 2
⇒ 2n – 4 = 2 × 360° ÷ 90 = 8
⇒ 2n – 4 = 8
⇒ 2n = 12 ⇒ n = 6
- The ratio between the number of sides of two regular polygons is 1 : 2 and the ratio between their interior angles is 2 : 3. The number of sides of these polygons is respectively
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We know that ,
Each interior angle = (2n - 4) × 90° n
According to question ,(2n - 4) × 90° n = 2 (4n - 4) × 90° 3 2n
[∵ Ratio of number of sides is 1 : 2]⇒ (2n - 4) × 2 = 2 4n - 4 3
Correct Option: C
We know that ,
Each interior angle = (2n - 4) × 90° n
According to question ,(2n - 4) × 90° n = 2 (4n - 4) × 90° 3 2n
[∵ Ratio of number of sides is 1 : 2]⇒ (2n - 4) × 2 = 2 4n - 4 3 ⇒ 2n - 4 = 1 4n - 4 3
⇒ 6n – 12 = 4n – 4
⇒ 6n – 4n = 12 – 4 = 8
⇒ 2n = 8 ⇒ n = 4
∴ No. of sides = 4, 8
- There are two regular polygons with number of sides equal to (n – 1) and (n + 2). Their exterior angles differ by 6°. The value of n is
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According to question ,
360° - 360° = 6° (n - 1) (n + 2) ⇒ 360° n + 2 - n + 1 = 6° (n - 1)(n + 2)
Correct Option: C
According to question ,
360° - 360° = 6° (n - 1) (n + 2) ⇒ 360° n + 2 - n + 1 = 6° (n - 1)(n + 2)
⇒ (n – 1) (n + 2) = 180°
n² + n – 2 = 180
n² + n – 182° = 0
n² + 14n – 13n – 182 = 0
n(n + 14) – 13 (n + 14) = 0
(n + 14) (n – 13) = 0
n = 13, – 14 [ ∴ n ≠ –14]
