if the angles of a concave polygon are 30,60,90,.......... then the number of diagonals of a polygon is
Answers
Given : angles of a concave polygon are 30,60,90, ........
To Find : the number of diagonals of a polygon is
Step-by-step explanation:
A concave polygon always have at least one reflex interior angle
an angle measuring between 180° and 360°
Let say polygon has n sides
Then sum of angles of a n sides polygon = (n - 2)(180°)
30,60,90, ..........,30n are the angles of n sides polygon
30n > 180 => n > 6
30 + 60 + 90 + .......+30n = (n - 2)(180)
Dividing both sides by 30
=> 1 + 2 + 3 +............+ n = (n - 2)6
∑n = (n)(n + 1)/2
=> n(n + 1)/2 = 6n - 12
=> n² + n =12n - 24
=> n² - 11n + 24 = 0
=> n² - 3n - 8n + 24 = 0
=> n(n - 3) - 8(n - 3) = 0
=> (n - 3)(n - 8) = 0
=> n = 3 , n = 8
n = 3 is not possible as its triangle and has no angle greater than 180°
Hence polygon has 8 sides
Number of diagonals in a polygon = n(n-3)/2
= 8 ( 8 - 3)/2
= 4 (5)
= 20
number of diagonals of polygon is 20
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Answer:
20
Step-by-step explanation: