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.
a. How far is the library from the Park?
b. How far is the park from the football field?
To find the distances, let's break down the information given step-by-step:
Given:
- 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.
a. How far is the library from the park?
To determine the distance between the library and the park, we need to use the Pythagorean theorem as we have a right triangle formed by the library, Katrina's home, and the park. Let's denote the distance between the library and the park as 'x'.
Using the Pythagorean theorem, we have:
(x^2) = (18^2) - (4^2)
(x^2) = 324 - 16
(x^2) = 308
x ≈ 17.55 miles
Therefore, the library is approximately 17.55 miles away from the park.
b. How far is the park from the football field?
Since the park is east of Katrina's home and the football field is south of the park, we can use the distance between Katrina's home and the football field along with the distance between the library and the football field to find the distance between the park and the football field.
The total distance between Katrina's home and the football field is the sum of the distance from her home to the library and the distance from the library to the football field.
Total distance: 4 miles (from her home to the library) + 18 miles (from the library to the football field) = 22 miles.
Therefore, the park is 22 miles away from the football field.
To solve this problem, let's break it down step by step:
a. How far is the library from the Park?
Step 1: Draw a diagram.
- Draw three points representing Katrina's home, the park, and the football field.
- Label Katrina's home as "K," the park as "P," and the football field as "F."
- Draw a line from Katrina's home to the football field, passing through the park.
- Label the point where the line intersects the line connecting Katrina's home and the park as "L" for the library.
Step 2: Determine the distances given in the problem.
- We know that the library is 4 miles from Katrina's home (KL = 4 miles).
- Also, the football field is 18 miles from the library (LF = 18 miles).
Step 3: Use the properties of right triangles to solve for the distance between the library and the park.
- Since there is a right triangle formed by Katrina's home, the park, and the football field, we can use the Pythagorean theorem to find the missing side length.
- The side from the library to Katrina's home is the hypotenuse of the right triangle.
- The legs of the right triangle are the distances from the park to the library (LP) and the park to Katrina's home (KP).
- Applying the Pythagorean theorem, we have:
KP^2 + LP^2 = KL^2
- We know that KL = 4 miles, and KP is the distance from Katrina's home to the park, but we need to determine LP, which is the distance from the library to the park.
Step 4: Solve for LP.
- Rearranging the equation from Step 3, we have:
LP^2 = KL^2 - KP^2
LP^2 = 4^2 - KP^2
LP^2 = 16 - KP^2
- To solve for LP, we need to find KP, which is the distance from Katrina's home to the park. The problem doesn't give us this information. Therefore, without additional information, we cannot determine the exact distance of the library from the park.
b. How far is the park from the football field?
- Based on the information given in the problem, we don't have enough information to determine the distance between the park and the football field.
using similar triangles, you can see that
PH/4 = 18/PH
PH^2 = 72
PH^2 + PF^2 = 18^2
PL/4 = PF/PH