Plane Geometry
- The ratio of each interior angle to each exterior angle of a regular polygon is 3 : 1. The number of sides of the polygon is
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Let Number of sides of polygon = n
According to the question,
The ratio of each interior angle to each exterior angle of a regular polygon = 3 : 1(2n - 4) 90° : 360° = 3 n n 1 ⇒ (2n - 4) × 90° = 3 360° 1 ⇒ 2n - 4 = 3 4 1
Correct Option: D
Let Number of sides of polygon = n
According to the question,
The ratio of each interior angle to each exterior angle of a regular polygon = 3 : 1(2n - 4) 90° : 360° = 3 n n 1 ⇒ (2n - 4) × 90° = 3 360° 1 ⇒ 2n - 4 = 3 4 1
⇒ 2n – 4 = 4 × 3 = 12
⇒ 2n = 12 + 4 = 16⇒ n = 16 = 8 2
- PQRST is a cyclic pentagon and PT is a diameter, then ∠PQR + ∠RST is equal to
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As per the given in question , we draw a figure cyclic pentagon PQRST
Sum of interior angles of a pentagon = (2n – 4) × 90°
Sum of interior angles of a pentagon = (2 × 5 – 4) × 90° = 540°
If PQ = QR = RS = ST∴ ∠POQ = ∠QOR = ∠ROS = ∠SOT = 180° = 45° 4
∴ OP = OQ = OR = OS = OT = radii∴ ∠OPQ = 180° - 45° = 135° 2 2
Correct Option: B
As per the given in question , we draw a figure cyclic pentagon PQRST
Sum of interior angles of a pentagon = (2n – 4) × 90°
Sum of interior angles of a pentagon = (2 × 5 – 4) × 90° = 540°
If PQ = QR = RS = ST∴ ∠POQ = ∠QOR = ∠ROS = ∠SOT = 180° = 45° 4
∴ OP = OQ = OR = OS = OT = radii∴ ∠OPQ = 180° - 45° = 135° 2 2 ∴ ∠PQR + ∠RST = 4 × 135° = 270° 2
- The interior angle of a regular polygon exceeds its exterior angle by 108°. The number of the sides of the polygon is
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Let the number of sides of regular polygon be n.
According to the question,
The interior angle of a regular polygon exceeds its exterior angle by 108°(2n - 4) × 90° - 360° = 108 n n ⇒ (2n - 4) × 5 - 20 = 6 n n
Correct Option: D
Let the number of sides of regular polygon be n.
According to the question,
The interior angle of a regular polygon exceeds its exterior angle by 108°(2n - 4) × 90° - 360° = 108 n n ⇒ (2n - 4) × 5 - 20 = 6 n n
⇒ 10n – 20 – 20 = 6n
⇒ 10n – 6n = 40
⇒ 4n = 40
⇒ n = 40 ÷ 4 = 10
- Measure of each interior angle of a regular hexagon is :
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As we know that ,
Each interior angle of regular polygon of n sides = 
2n - 4 
× 90° n
Here, n = 6∴ Required answer = 2 × 6 - 4 × 90° 6
Correct Option: D
As we know that ,
Each interior angle of regular polygon of n sides = 2n - 4 × 90° n
Here, n = 6∴ Required answer = 2 × 6 - 4 × 90° 6 Required answer = 8 × 90° = 120° 6
- If the sum of all interior angles of a regular polygon is 14 right angles, then its number of sides is
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We know that Sum of all interior angles of a regular polygon of n sides = (2n – 4) right angles
According to question ,
∴ 2n – 4 = 14
⇒ 2n = 14 + 4 = 18Correct Option: B
We know that Sum of all interior angles of a regular polygon of n sides = (2n – 4) right angles
According to question ,
∴ 2n – 4 = 14
⇒ 2n = 14 + 4 = 18⇒ n = 18 = 9 2
