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 4 miles from her home. The football field is 10 miles from the library.
How far is the library from the park?
How far is the park from the football field
A. 7 miles and 12 miles
B.2 square root of 10 and 2 square root of 6
C.square root of 14 and 2 square root of 6
D. 2 square root of 10 and 2 square root of 35
Label the various points P,K,F,L
using similar triangles,
PL/4 = 10/PL
PL=2√10
10/PF=PF/14
PF=2√35
To find the distance between the library and the park, we can use the Pythagorean theorem. Let's assume that the distance between Kristen's home and the park is 'x' miles.
According to the information given, the library is 4 miles from Kristen's home, and the football field is 10 miles from the library. We can consider the distance between the library and the park as the base of a right triangle, the distance between Kristen's home and the park as one of the legs, and the distance between Kristen's home and the football field as the hypotenuse.
Using the Pythagorean theorem, we have:
(x^2) + (4^2) = (10^2)
Simplifying this equation:
x^2 + 16 = 100
x^2 = 100 - 16
x^2 = 84
Taking the square root of both sides:
x = √84
Simplifying the square root of 84:
x ≈ 2√21
So, the distance between the library and the park is approximately 2√21 miles.
Next, to find the distance between the park and the football field, we can subtract the distance between Kristen's home and the library (4 miles) from the distance between Kristen's home and the football field (10 miles):
10 - 4 = 6
Therefore, the park is 6 miles from the football field.
Thus, the answer is B. 2 square root of 10 and 2 square root of 6.