. Katrina lives directly east of the park. The football field is directly south of the park. The library sits on the line formed between Katrina’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 18 miles from the library.
I don't see any question here, but if you label the triangle's vertices as
PKF, with P as the right angle, then the library sits at L, where
∆PKF ~ ∆LFP ~ ∆LKP
You can get the length of PL or FL or KL from those ratios.
To determine the distance between Katrina's home and the park, we can draw a diagram based on the given information.
Let's assume that Katrina's home is point H, the park is point P, the football field is point F, and the library is point L.
Based on the description, we know that Katrina's home is directly east of the park, meaning that the line segment HP is a horizontal line. The football field is directly south of the park, so we can draw a vertical line segment PF.
Now, we are told that the library sits on the line formed between Katrina'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 line segment HL represents the altitude of the right triangle.
Given that the library is 4 miles from Katrina's home and the football field is 18 miles from the library, we can determine the remaining side lengths of the right triangle.
Using the Pythagorean theorem, we can find the distance between Katrina's home and the park (HP). Let's call this distance x.
Based on the right triangle formed by HP, HL, and PL, we have the following equation:
x^2 + 4^2 = 18^2
Simplifying this equation, we get:
x^2 + 16 = 324
Subtracting 16 from both sides, we have:
x^2 = 308
Taking the square root of both sides, we find:
x ≈ √308
x ≈ 17.58
Therefore, the distance between Katrina's home and the park (HP) is approximately 17.58 miles.
To solve this problem, let's break it down step-by-step:
Step 1: Identify the given information:
- Katrina lives directly east of the park.
- The football field is directly south of the park.
- The library sits on the line formed between Katrina's home and the football field.
- The library is 4 miles from her home.
- The football field is 18 miles from the library.
Step 2: Draw a diagram:
Park
|
Katrina -- Library -- Football Field
Step 3: Use the information to determine the distance between the park and the library:
Since the library is located on the line formed between Katrina's home and the football field, and it is 4 miles from her home, we can conclude that the distance between the park and the library is also 4 miles.
Step 4: Determine the distance between Katrina's home and the football field:
Since the football field is 18 miles from the library, and the library is 4 miles from Katrina's home, we can subtract these two distances to find the distance between her home and the football field. Subtracting 4 miles from 18 miles gives us a distance of 14 miles.
Step 5: Summarize the information:
- Katrina's home is directly east of the park.
- The distance between the park and the library is 4 miles.
- The distance between the library and the football field is 18 miles.
- The distance between Katrina's home and the football field is 14 miles.