Problem: (2003 AIME1 – #12) In convex quadrilateral , , , and . The perimeter of is . Find .
Solution: Well, let’s say that and . We know . Consider the two triangles and . Using the Law of Cosines on and , which we know are equal (let them both equal ), we have
Substituting and , , it becomes
Equating the two,
Recall that , so
Finally, since , we divide through to get
Comment: This is a demonstration of the power of something as simple as the Law of Cosines. Never underestimate what a little experimentation can do for you on the AIME; play around with equations. If at first you do not see an approach, look at the question itself for hints. Since you have to find the cosine, it should immediately trigger the Law of Cosines because it relates sides and angles. Then apply the Law of Cosines to the important angles (the ones you know something about), in this case and , from which the result falls quite nicely.
Practice Problem: (2003 AIMEI – #6) The sum of the areas of all triangles whose vertices are also vertices of a cube is , where , , and are integers. Find .
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