A regular pentagon is inscribed in a circle. If A and B are

Question

A regular pentagon is inscribed in a circle. If A and B are adjacent vertices of the pentagon and O is the center of the circle, what is the value of ∠OAB ?

A. 48 degrees 
B. 54 degrees 
C. 72 degrees 
D. 84 degrees 
E. 108 degrees

Marcab · Accepted Answer

Since it is mentioned that the pentagon is regular, hence its all the sides are equal. In the diagram attached, if you join the vertices of the pentagon with the centre, then you will get 4 isosceles triangle. Each triangle will have a central angle i.e. angle at the centre of the circle and two equal angles, both at the vertex of the pentagon.
So there will be 5 central angles. Name them as x. So as per the property of the circle, the sum of the entire central angle will be 360.
Hence  5x=360  or  x=72 .
Its now clear that the value of central angle of each triangle will be 72. Therefore the sum of the rest of the angles will be  180-72=108 
 108  is the sum of two equal angles, hence each angle will be equal to  54 . 
The value of the angle OAB is 54.

Fistail · Answer

its a regular pentagon so divide the globe by 5. 
= 360/5
= 72

sonibubu · Answer

72 would be the angle at O.
Angles at A and B are equivalent, so

72 + 2x = 180
2x = 108
x = 54

Angles OAB and OBA will be 54 degrees each.
B

Fistail · Answer

oh, its the edge not the center. ok, lol.

ywilfred · Answer

OA and OB are radius of a circle, so we have an isoceles triangle. Also,360degrees in the centre of circle will be divided up equally into 5 parts, each measuring 360/5 = 72 degrees. 

So 72 + 2x = 180 where x = OAB = OBA

x = 54

bmwhype2 · Answer

OA is B 54.

1/5 of 360 is 72. 

180 - 72 = 102

102/2 isos angles = 54

pikachu · Answer

Hey marcab gr8 explanation. Thx

CIyer · Answer

Another way of cracking this sum: the sum of angles of a polygon is 180 (n-2). n stands for the no. of sides of the polygon. So, here we have n=5. Therefore, sum of all angles= 180 (5-2)= 540. Since, this is a regular figure each angle= 540/5= 108. However, 108 is the complete angle. Thus, half the angle = 54. (The angle with the center of circle/point of intersection of all diagonals will always be 0.5 times of the complete angle in all regular figures).

Jaisri · Answer

How do we assure that the radius is bisecting the angle? Isn't there a possibility that the radius will cut the angles into 2 unequal parts?

Do you mean for all regular polygons? Because if the figure is a regular hexagon, then the angle with the center will be 0.6 times the complete angle. Kindly clarify.

CIyer · Answer

from a logical standpoint, in all regular figures the diagonals (drawn through the centre) will bisect the angles. Consider a regular hexagon ABCDEF. let O be the point of intersection of all diagonals. Here angle ABC= angle(OBA) + angle (OBC). angle (OBA) = angle (OBC) = 60. Thus, regular hexagon is made of 6 equilateral triangles. Hope it is clear. Perhaps, Bunnel can explain this in a better manner.

skamal7 · Answer

I used the formula the sum of interior angles of pentagon 180(n-2) which gives 540 since pentagon 540/5=108 So this angle is <AOB right? And by using central angle theorem can we deduce that the answer is<OAB =108/2= 54? Is my understanding on this problem corrrect??can you please verify
?

Zarrolou · Answer

Nope. 540° is the sum of the angles of the pentagon, as you correctly say, so angles A, B would be 108, not angle AOB.

Angle <AOB is not  \frac{540}{5}  but it's  \frac{360}{5}  (a circle divided into 5 parts).

According to your calculus angle AOB is 108 and the other two angles are 54: that makes a total of a 216° triangle...

I solved like this:
Angle AOB is  72 , and is part of an isosceles triangle, so the other two angles will be equal:
 180-72=108  and each angle is  \frac{108}{2}=54  ( 72+54+54=180 )

skamal7 · Answer

So for pentagon what will be the central angle and interior angle?

Zarrolou · Answer

The central angle is 360° (it's a circle).
The interior angles (of a regular pentagon) are  \frac{540}{5}=108  each.

Jaisri · Answer

It makes sense as it matches the solution derived using another approach (isoceles triangle). Just wanted to check if there is any property that I should be aware of. Thanks for the explanation!

kalrac · Answer

The sum of interior angles of n sides :  (n-2)*180  , here its 540 
Any polygon inscribed in a circle is  Regular polygon  implies all the angles are equal , hence each angle is 540/5 = 108
OAB = 108/2 = 54.

jlgdr · Answer

Totally agree with explanation given by kalrac Just to add there is other way Find the central angle, that would be 360/5 = 72 Then 2x + 72 = 180 So x is also 54 Hope it helps! Cheers! J

rrsnathan · Answer

Understood the question wrongly and marked the angle as 72 which is the angle from center.

saurabhprashar · Answer

A regular pentagon is inscribed in a circle. If A and B are

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