Question 1209330
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The AI-produced "solution" to the problem is incorrect.<br>
If you try to follow it, you will see that it says segment CF is a side of the heptagon, which it is not.<br>
You should also recognize that, with the side length of the heptagon being 1, it is absurd that the answer would be that the radius of the circle tangent to DC at C and to EF at F is less than 1/2....<br>
Following is a correct solution to the problem.  In that solution, I only outline the calculations, and I only show numbers to a few decimal places.  In order to learn anything from the problem, the student should go through the detailed calculations himself (I used a TI-84 calculator), and he should not do any rounding until the final answer.<br>
Let O be the center of the circle.  Then, according to the given information, OC is a radius of the circle and is perpendicular to CD, and OF is a radius of the circle perpendicular to FE.<br>
Each exterior angle of the heptagon has a measure of 360/7 degrees, so each interior angle has a measure of 180 - (360/7) = 900/7 degrees.<br>
Use the law of cosines in triangle DEF with DE=EF=1 and angle E having a measure of 900/7 degrees to find that the length of DF is about 1.8018.<br>
In isosceles triangle DEF, angle DEF has measure 900/7 degrees, so each of the other two angles has measure (1/2) of (180 - (900/7)) = 180/7 degrees.  That makes the measure of angle CDF 900/7 - 180/7 = 720/7 degrees.<br>
Use the law of cosines in triangle CDF, knowing the lengths of CD and DF and the measure of angle CDF, to find that the length of CF is about 2.24698.<br>
In pentagon OCDEF, use the known measures of angles OCD, CDE, DEF, and EFO to determine that the measure of angle COF is 720/7 degrees.  Then use the law of cosines in triangle OCF, with OC and OF being radii of the circle, to find that the radius of the circle is about 1.437.<br>
ANSWER (approximately): 1.437<br>