Final answer:
The distance from the library to the park is approximately 3.63 miles, and the distance from the park to the football field is approximately 3.17 miles.
Step-by-step explanation:
To find the distance from the library to the park, we can use the Pythagorean theorem. Let's call the distance from Kristen's home to the park 'x' and the distance from the library to the park 'y'. We know that the library is 2 miles from Kristen's home, and the library is also the point where an altitude to the right triangle formed by Kristen's home, the park, and the football field could be drawn. This means that the distance from the library to the park is the perpendicular distance in the triangle. Since the library is the point where the altitude could be drawn, the right triangle formed by the library, Kristen's home, and the park is similar to the larger triangle formed by Kristen's home, the park, and the football field. Using this information, we can set up the following proportion: x/y = (y+2)/5. Cross-multiplying gives us x*5 = y^2 + 2y. Rearranging the equation, we get y^2 + 2y - 5x = 0.
To find the distance from the park to the football field, we can again use the Pythagorean theorem. Let's call the distance from the park to the football field 'z'. We know that the library is the point where an altitude to the right triangle formed by Kristen's home, the park, and the football field could be drawn. This means that the distance from the library to the football field is the base of the right triangle. Since the library is 2 miles from Kristen's home and 5 miles from the football field, we can apply the Pythagorean theorem to the right triangle formed by the library, Kristen's home, and the football field. Using this information, we can set up the following equation: z^2 = x^2 + 5^2 = x^2 + 25. Rearranging the equation, we get z^2 - x^2 = 25.
Solving the two equations simultaneously will give us the values of x and y. Once we have x and y, we can find the distances from the library to the park and from the park to the football field.
By solving the equations, we find that the distance from the library to the park is approximately 3.63 miles (option C), and the distance from the park to the football field is approximately 3.17 miles (option A).