We can use a theorem called Proportional segments in right triangles.
1. How far is library from the park?
The perpendicular is the mean proportional between the segments of the hypotenuse, so:
[tex]PL= \sqrt{FL \cdot HL } = \sqrt{8 \cdot 2 } = \sqrt{16}=4 \ miles [/tex]
2. How far is the park from the football field?
Each leg of the given triangle is the mean proportional between the hypotenuse and the adjacent segment, so:
[tex]PF= \sqrt{FH \cdot FL } = \sqrt{(8+2) \cdot 8 } = \sqrt{80}=4 \sqrt{5} \ miles [/tex]
Answer: A.
