2007 Alabama ARML TST Problems/Problem 15
Problem
Let be a point inside isosceles right triangle such that , , , and . Find the area of .
Solution
Solution 1
Let , , and be the reflections of over sides , , and , respectively. We then have that , , and . This shows that . I shall now proceed to find . This is equal to
Note that and , so . Similarly, and . Now note that and . Therefore and . Also note that and . We also know that , , and are collinear, so . This shows that is a 5-12-13 right triangle, so it has area , so
Solution 2
Rotate the diagram by 90 degrees about so that goes to , goes to a point , and goes to . Since the image of under this rotation is , . Since is a 45-45-90 right triangle, . Thus, is a 5-12-13 right triangle, with . Note that , so by Law of Cosines on triangle ,
so .
See also
2007 Alabama ARML TST (Problems) | ||
Preceded by: Problem 14 |
Followed by: Last Problem | |
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