Kristen lives directly east of the park. The football field is directly south of the park. The library sits on the line formed between Kristen’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 2 miles from her home. The football field is 8 miles from the library.

How far is library from the park?

How far is the park from the football field?
A. 4 miles; miles
B. miles; miles
C. 5 miles; 9 miles
D. 4 miles; miles

Trying to match the question with the choices sure stumps me.

Using similar triangles, if we label the vertices of the triangle P,K,F, and the library L, then we have

∆LPF ~ ∆KLP
we want the distance LP and FP, and we have

LP/2 = FP/KP
LP/8 = KP/FP
so,

LP/2 = 8/LP
LP^2 = 16
LP = 4

FP^2 = LP^2 + LF^2 = 4^2+8^2 = 80
FP = 4√5

To find the distance between the library and the park, we need to consider the right triangle formed by Kristen's home, the park, and the football field.

Let's assume that the library is point L, Kristen's home is point K, the park is point P, and the football field is point F.

We know that Kristen's home is directly east of the park and the library sits on the line formed between Kristen’s home and the football field. Thus, the line segment KL represents the direction from Kristen's home towards the park.

Since the library is at the exact point where an altitude to the right triangle can be drawn, the line segment LP represents the altitude.

Given that distance KL is 2 miles, and distance LF is 8 miles, we can use the Pythagorean theorem to find the distance LP.

According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (in this case, LF) is equal to the sum of the squares of the other two sides (in this case, KL and LP).

Using the Pythagorean theorem, we get:

LF^2 = KL^2 + LP^2

(8 miles)^2 = (2 miles)^2 + LP^2

64 miles^2 = 4 miles^2 + LP^2

60 miles^2 = LP^2

Taking the square root of both sides, we get:

LP = √(60) miles

So, the distance between the library and the park is approximately √(60) miles, which is around 7.75 miles.

To find the distance between the park and the football field, we can use the distance LF, which is 8 miles.

Therefore, the distance between the library and the park is approximately 7.75 miles, and the distance between the park and the football field is 8 miles.

The answer is: C. 7.75 miles; 8 miles.

To find the distance between the library and the park, we need to determine the length of the line formed between Kristen's home (which is directly east of the park) and the library.

According to the information given, the library sits on the line formed between Kristen’s home, the park, 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.

This means that the line from Kristen's home to the library is the hypothenuse of a right triangle, and the distance between the library and Kristen's home is the length of this hypothenuse.

We already know that the distance between Kristen's home and the football field is 8 miles, and the distance between the library and Kristen's home is 2 miles. By using the Pythagorean theorem (a^2 + b^2 = c^2), we can find the length of the hypothenuse.

Let's calculate it:
a = 2 miles (distance between library and Kristen's home)
b = 8 miles (distance between football field and Kristen's home)

c^2 = a^2 + b^2
c^2 = 2^2 + 8^2
c^2 = 4 + 64
c^2 = 68
c ≈ 8.246 miles

So, the distance between the library and the park is approximately 8.246 miles.

As for the distance between the park and the football field, we can use the information given. Since Kristen's home is directly east of the park and the football field is directly south of the park, we can conclude that the park is located at the intersection of the line from Kristen's home to the library and the line from the library to the football field.

Therefore, the distance between the park and the football field is the length of the line connecting them, which is given as 4 miles (option A).

In summary:
- The library is approximately 8.246 miles from the park.
- The park is 4 miles from the football field.