1971 AHSME Problems/Problem 26
Contents
[hide]Problem
In , point divides side in the ratio . Let be the point of intersection of side and where is the midpoint of . The point divides side in the ratio
Solution 1
We will use mass points to solve this problem. is in the ratio so we will assign a mass of to point a mass of to point and a mass of to point
We also know that is the midpoint of so has a mass of so also has a mass of
In line has a mass of and has a mass of Therefore,
The answer is
-edited by coolmath34
Solution 2
By Menelaus' Theorem on and line , we know that . Because is the midpoint of , we know that . Furthermore, from the problem, we know , so, by susbtitution, our first equation becomes , so , and therefore .
Using Menelaus again on and line reveals that . We know that and . Thus, our equation becomes , and so . Because , we see that , so our answer is .
See Also
1971 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 25 |
Followed by Problem 27 | |
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