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Let $$AB$$ be a line segment of length 26. Let $$C$$ and $$D$$ be located on the line segment $$AB$$ such that $$AC = 1$$ and $$AD = 8$$. Let $$E$$ and $$F$$ be the points on one of the semi circles with diameter $$AB$$ for which $$EC$$ and $$FD$$ are perpendicular to $$AB$$. Find the length of the line segment $$EF$$.
Correct Answer: 9.9
For a point on a semicircle, the perpendicular to the diameter satisfies $$h^2 = $$ (product of the two parts of the diameter). So $$EC^2 = 1 \times 25 = 25$$, giving $$EC = 5$$, and $$FD^2 = 8 \times 18 = 144$$, giving $$FD = 12$$. Then $$EF = \sqrt{(8 - 1)^2 + (12 - 5)^2} = \sqrt{98} = 7\sqrt{2} \approx 9.9$$.
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