Find the Distance

Geometry Level 4

Let ΔABC \Delta ABC be the right triangle with side lengths AB=15AB = 15, BC=8 BC = 8, and AC=17AC = 17 . Let DD be the point on the segment AC\overline{AC} between AA and CC whose distance to AA is 1, and let EE be the point on the segment AB\overline{AB} between AA and BB whose distance to AA is 3. Let FF be the point of intersection of the lines DE\overleftrightarrow{DE} and BC\overleftrightarrow{BC}. The distance BFBF can be written as ab \frac{a}{b} , where aa and bb are coprime positive integers. What is the value of a+ba+b?

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