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 6 miles from h
er home. The football field is 8 miles
from the library.
A. √14 miles; √22 miles
B. 7 miles; 11 miles
C. 4√3 miles; 4√7 miles
D. 4√3 miles; √22 miles
I think it's D
To determine Kristen's location relative to the park, library, and football field, let's analyze the given information step-by-step.
1. Kristen lives directly east of the park.
- This means that her home is located to the right (east) of the park.
2. The football field is directly south of the park.
- This indicates that the football field is located below (south of) the park.
3. 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.
- This means that the library is located on the line connecting Kristen's home and the football field. Additionally, this line is perpendicular to the line connecting her home and the park, forming a right triangle.
4. The library is 6 miles from her home.
- This tells us that the distance between Kristen's home and the library is 6 miles.
5. The football field is 8 miles from the library.
- This indicates that the distance between the library and the football field is 8 miles.
Based on this information, we can infer the following:
- Kristen's home is directly east of the park.
- The football field is directly south of the park.
- The library is located on the line connecting Kristen's home and the football field.
- Kristen's home, the park, and the football field form a right triangle, with the library marking the point where an altitude could be drawn.
Please let me know if you need clarification on any of the steps or if you have any other questions.
To find the distance between Kristen's home and the park, we need to use the information given in the question. 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 at the point where an altitude to the right triangle formed by her home, the park, and the football field could be drawn.
Let's assign some labels to the given points:
- Kristen's home: A
- The park: B
- The library: C
- The football field: D
Since Kristen's home is directly east of the park, we can draw a horizontal line connecting A and B. Suppose this line intersects the line CD (which is the line formed between Kristen's home and the football field) at point E.
Now, we have a right triangle ABE, with AE representing the distance between Kristen's home and the park, and AB representing the distance between Kristen's home and the library. We also know that CD is the altitude to this right triangle at point E.
According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, AE and AB are the two sides, and CD is the hypotenuse.
Let's use this information to set up an equation:
AB^2 + AE^2 = CD^2
We are given that AB = 6 miles and CD = 8 miles. Plugging these values into the equation:
6^2 + AE^2 = 8^2
36 + AE^2 = 64
AE^2 = 64 - 36
AE^2 = 28
To find AE, we take the square root of both sides:
√(AE^2) = √28
AE = √28
AE ≈ 5.29 miles
So, the distance between Kristen's home and the park (AE) is approximately 5.29 miles.