Mock AIME 3 Pre 2005 Problems/Problem 9
Problem
is an isosceles triangle with base . is a point on and is the point on the extension of past such that is right. If and , then can be expressed as , where and are relatively prime positive integers. Determine .
Solution 1
Let AB=x. Call the foot of the perpendicular from D to AB N, and the foot of the perpendicular from C to AB M. By similarity, AN=2x/17. Also, AM=x/2. Since AND and CAM are similar, we have (2x/17)/AD=(x/2)/16. Hence, AD=64/17, and CD=16-AD=208/17, so the answer is 225.
Solution 2 (Mass points)
Let the perpendicular from intersect at Let intersect at Then let intersect at
Note that with a factor of So and Then the mass of is and the mass of is and the mass of is Because the triangle is isosceles, the mass of is also So
See Also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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