To find the distance between Kathy's home and the park, we need to first visualize the situation. Let's call the distance between Kathy's home and the park x, and the distance between the park and the football field y.
Since the library sits on the line formed between Kathy's home and the football field at the exact point where an altitude to the right triangle formed by her home, the park, and the football field could be drawn, we can create a right triangle with Kathy's home, the park, and the football field as the vertices. The length of the altitude from the park to the line segment between Kathy's home and the football field is 9 miles (the distance between Kathy's home and the library).
This right triangle can be split into two smaller right triangles, one with sides x, y, and 9 (the distance between Kathy's home and the library), and the other with sides (x + 12), y, and 9.
Using the Pythagorean Theorem for both triangles, we get:
x^2 + y^2 = 9^2
and
(x + 12)^2 + y^2 = 9^2
Solving these two equations simultaneously, we get:
x^2 + y^2 = 81
(x + 12)^2 + y^2 = 81
Expanding and simplifying the second equation, we get:
x^2 + 24x + 144 + y^2 = 81
Substitute the x^2 + y^2 = 81 from the first equation:
24x + 144 = 0
24x = -144
x = -6
Since x represents the distance between Kathy's home and the park and distance cannot be negative, there seems to be an error in the given information or in the assumptions made.
Kathy lives directly east of the park. The football field is directly south of the park. The library sits on the line formed between Kathy’s home and the football field at the exact point where an altitude to the right triangle formed by her home, the park, and the football field could be drawn. The library is 9 miles from her home. The football field is 12 miles from the library.
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