Difference between revisions of "2010 AMC 10A Problems/Problem 19"
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Revision as of 04:11, 16 January 2020
Equiangular hexagon has side lengths and . The area of is of the area of the hexagon. What is the sum of all possible values of ?
this diagram only shows one possible value of r
It is clear that is an equilateral triangle. From the Law of Cosines on triangle ABC, we get that . Therefore, the area of is .
If we extend , and so that and meet at , and meet at , and and meet at , we find that hexagon is formed by taking equilateral triangle of side length and removing three equilateral triangles, , and , of side length . The area of is therefore
Based on the initial conditions,
Simplifying this gives us . By Vieta's Formulas (or Girard identities, or Newton-Girard identities) we know that the sum of the possible value of is .
As above, we find that the area of is .
We also find by the sine triangle area formula that , and thus This simplifies to .
Solution 3 (no trig)
Extend to point M so that it creates right triangle where . It is given that the hexagon is equiangular, therefore . (exterior angles of a polygon add up to 360 )
We can use either Pythagorean theorem or the properties of a triangle to find the length of and . The legs of are and .
Using Pythagorean theorem, we get . We can then follow to solve for . .
Alternatively, we can find the area of . We know that the three smaller triangles: , , and are congruent because of . Therefore one of the smaller triangles accounts for of the total area. The height of the smaller triangle is just so the area is . We can then find the area of the hexagon using .
We can even find the area of and and solve for because the ratio of the areas is to .
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