2006 iTest Problems/Problem 27
Problem
Line passes through and into the interior of the equilateral triangle . and are the orthogonal projections of and onto respectively. If and , then the area of can be expressed as , where and are positive integers and is not divisible by the square of any prime. Determine .
Solution
Let be the intercept of and . By the Vertical Angle Theorem, . Also, since both and are perpendicular to , . Thus, by AA Similarity. Since , and .
Let be the side length of the triangle, so . By the Pythagorean Theorem, . Also, , so by the Law of Cosines, .
By using the Pythagorean Theorem again, we have Thus, the area of the triangle is , so .
See Also
2006 iTest (Problems) | ||
Preceded by: Problem 26 |
Followed by: Problem 28 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • U1 • U2 • U3 • U4 • U5 • U6 • U7 • U8 • U9 • U10 |