SOLUTION: I have a circle inside a regular pentagon. CE is from top of pentagon to the bottom. I know that CE is 1.4375. I need to find DE which is the radius of the circle. CE is a straigh

Question 391000: I have a circle inside a regular pentagon. CE is from top of pentagon to the bottom. I know that CE is 1.4375. I need to find DE which is the radius of the circle. CE is a straight line going all the way through middle of the circle.
Found 2 solutions by scott8148, Edwin McCravy:
Answer by scott8148(6628) (Show Source): You can put this solution on YOUR website!
picture the pentagon divided into ten congruent right triangles
___ by lines drawn from the midpoint of a side to the opposite vertex

the "central" angle of the right triangle (in the center of the pentagon) is 36� (360� / 10)

the length of the side of the triangle from the right angle to the center of the pentagon is D/2
___ this same side , added to the hypotenuse of the right triangle , equals C

(D / 2) + h = C

from trigonometry ___ (D / 2) / h = cos(36�) ___ (D / 2) / cos(36�) = h

substituting ___ (D / 2) + [(D / 2) / cos(36�)] = C

D{1 + [1 / cos(36�)]} = 2C

D = 2C / {1 + [1 / cos(36�)]}
Answer by Edwin McCravy(20085) (Show Source): You can put this solution on YOUR website!
I have a circle inside a regular pentagon. CE is from top of pentagon to the bottom. I know that CE is 1.4375. I need to find DE which is the radius of the circle. CE is a straight line going all the way through middle of the circle.

Draw DA and DB: Angle ADC = = Angle BDA = = Therefore angle BDC = Erase DA (to keep figure from being cluttered) Draw the diagonal CB, and label angle BDC as having measure 144� Since triangle CDB is isosceles and has vertex angle 144�, its two congruent base angles are 36� each because 180�-144�= 36� and each of the base angles is one-half of that. So we label angle BCD as 18�: CEB is a right triangle. Therefore =tan(18�) BE = CE*tan(18�) and since CE is given as 1.4375 BE = 1.4375*tan(18�) = 0.4670720633 Angle BDE is suplementary to angle CDB which 144�, so angle BDE is 180�-144� = 36�. Now I'll erase some more: Now triangle DBE is a right triangle, so = sin(36�) BE = DB*sin(36�) DB = = = 0.7946304565. Since DB is a radius of the circle, that's what was required. Answer radius = 0.7946 rounded to nearest ten thousandth. Edwin

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