SOLUTION: A regular pentagon has sides of 10cm. Find the radius the largest circle which can be drawn inside the pentagon. Actually this question is from the chapter of Triginometry: Tang

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Question 207274This question is from textbook Longman Mathematics for IGCSE book 1
: A regular pentagon has sides of 10cm.
Find the radius the largest circle which can be drawn inside the pentagon.
Actually this question is from the chapter of Triginometry: Tangent ratios
Please tell me that how could I use tangent in this particular question.
I request you to solve my question as soon as possible. I would be highly glad. Thankyou very much indeed. This question is from textbook Longman Mathematics for IGCSE book 1

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
the circle inscribed in the pentagon will touch each side of the pentagon exactly in the middle and be perpendicular to the side of the pentagon at that point.
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see the diagrram at the following website to see what i mean:
http://etc.usf.edu/clipart/36700/36715/pentcirc_36715.htm
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you may also go to this website and click on problem number 207274 to get another picture with some additional information.
http://theo.x10hosting.com/
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the answer to your problem is as follows:
central angles of a pentagon are each 72 degrees (360 / 5 = 72)
line drawn from the center of one of the triangles within the pentagon to the side of the pentagon cuts the side of the pentagon in half and is perpendicular to it.
this divides one of the triangle of the pentagon into 2 right triangles.
the central angle of one of the right triangle is 72 / 2 = 36 degrees.
the line bisecting the side of the triangle intersects the side of the pentagon at 90 degrees.
this means the other angle of the right triangle is 180 - 90 - 36 = 54 degrees.
to find the length of the bisector of the side of the pentagon we find the tangent of 54 degrees.
the tangent of 54 degrees = opposite side over adjacent side.
the adjacent side is 5 cm.
the opposite side is unknown.
the formula is tan (54) = o/5.
this becomes 1.37638192 = o/5
solving for o we get o = 1.37638192 * 5 = 6.881909602
this turns out to be the radius of the inscribed circle.
the inscribe circle is the largest circle that can be inscribed within the pentagon.
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your answer is radius of largest circle that can be inscribed within the pentagon is 6.881909602
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check the website of http://theo.x10hosting.com/ and click on problem 207274 to see how this works. if it's not there when you check, wait 1/2 hour and check again. it will be there.
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in the diagram on that website:
A = circumscribed circle (outside of pentagon)
B = inscribed circle (inside of pentagon)
D is the cenhter of the circle and the center of the pentagon.
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the pentagon is a regular pentagon meaning all angles are equal and all sides are equal. each central angle of the pentagon is 72 degrees (5 * 72 = 360)
total angles of the pentagon = 540 degrees. this makes each vertex of the pentagon equal to 108 degrees. if you draw lines from the center of the pentagon to each vertex, these lines divide the vertex angles into 2 54 degree angles each.
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DG is perpendicular to side EF of the pentagon at point G.
DG is perpendicular to EF at G
EF is 10 centimeters long
DG bisects EF at G so EG and GF are each 5 centimeters long.
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triangle DGF is a right triangle.
angle GFD is 54 degrees (half of 108 degrees)
tan (54) = opposite divided by adjacent = DG / GF = DG / 5
solve for DG gets DG = 5 * tan(54) = 5 * 1.37638192 = 6.881909602
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EF is tangent to the circle at G which means it is perpendicular to the radius DG of the circle at that point.
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