Christian lives directly east of the park the football field is directly south of the park the library sits on the line phone between Kristen’s home and the football field at that exact point where in altitude to the right triangle formed by her home the park and the football field could be drawn the library is 2 miles away from her home the football field is 5 miles away from the library . How far is the library from the park? How far is the Park from the football field?
To solve this problem, we will use the concept of the Pythagorean theorem and the properties of right triangles.
Let's start by drawing a diagram to visualize the information given. We will label Christian's home as point C, the park as point P, the library as point L, and the football field as point F.
Now, we are given that the library sits on the line between Christian's home (C) and the football field (F) at a specific point. This point is exactly where the altitude of the right triangle formed by her home (C), the park (P), and the football field (F) could be drawn. Let's label this point as A.
Next, let's focus on the given distances:
- The library (L) is 2 miles away from Christian's home (C).
- The football field (F) is 5 miles away from the library (L).
Now, notice that the right triangle formed by the points C, P, and A is similar to the right triangle formed by the points L, P, and A. This is because both triangles share an angle at point P and have a right angle at point A.
Using this similarity, we can set up a proportion to find the distance between the library (L) and the park (P).
Let x be the distance between the library (L) and the park (P). Then, we can set up the following proportion:
x/2 = 5/3
To solve for x, we can cross-multiply:
3x = 2 * 5
3x = 10
x = 10/3
Therefore, the library is located approximately 3.33 miles away from the park.
To find the distance between the park (P) and the football field (F), we can use the same approach. Since the distance between the library (L) and the football field (F) is 5 miles, we can set up the following proportion:
5/x = 3/2
Cross-multiplying:
2 * 5 = 3x
10 = 3x
x = 10/3
Therefore, the park is approximately 3.33 miles away from the football field.
So, the library is approximately 3.33 miles away from the park, and the park is approximately 3.33 miles away from the football field.
As always, draw a diagram. If we label the locations with the obvious letters (P,H,F,L) then we have
∆FPH ~ ∆FLP ~ ∆PLH
We want LP and PF. Using similar triangles,
5/PF = PF/7
PF² = 35
LP²+FL² = PF²
LP² + 25 = 35
LP² = 10