1967 AHSME Problems/Problem 32
Problem
In quadrilateral with diagonals and , intersecting at , , , , , and . The length of is:
Solution 1
After drawing the diagram, we see that we actually have a lot of lengths to work with. Considering triangle ABD, we know values of , but we want to find the value of . We can apply stewart's theorem now, letting , and we have , and we see that
Solution 2
(Diagram not to scale)
Since , is cyclic through power of a point. From the given information, we see that and . Hence, we can find and . Letting be , we can use Ptolemy's to get Since we are solving for
- PhunsukhWangdu
Solution 3 (Law of Cosines Cheese)
The solution says it all. Since is supplementary to , . The law of cosines on gives us . Again, we can use the law of cosines on , which gives us which gives us
Note that this solution works even if the quadrilateral is not cyclic, and in general, it works if an angle's supplement is known. (Anyone come from aops volume 2 lmao.)
-Wesssslili
See also
1967 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 31 |
Followed by Problem 33 | |
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