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 her home. The football field is 8 miles from the library.
How far is the library from the park? How far is the park from the football field?
A.√22 miles; 2√10 miles
B. 2√10 miles; √22 miles
C. 6√2 miles; 6√11 miles
D. 6√11 miles; 6√2 miles
To find the distances between the library, park, and football field, let's break down the problem step by step.
Step 1: Draw a diagram
To visualize the situation, draw a diagram with Kristen's home marked as a point east of the park, the park itself, and the football field directly south of the park. Then, mark the library on the line connecting Kristen's home and the football field.
Step 2: Identify the right triangle
The problem mentions a right triangle formed by Kristen's home, the park, and the football field. We need to find the side lengths of this triangle.
Step 3: Label the known side lengths
We are given that the library is 6 miles from Kristen's home and the football field is 8 miles from the library.
Step 4: Use the Pythagorean theorem
Since we have a right triangle, we can use the Pythagorean theorem to find the length of the remaining sides.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In our case, let's call the distance between the library and the park "x" and the distance between the park and the football field "y." Now we can set up the equation:
(6^2) + (x^2) = (8^2) -- equation 1
(x^2) + (y^2) = (y^2) -- equation 2
Step 5: Solve the equations
Simplifying equation 1:
36 + (x^2) = 64
(x^2) = 28
Taking the square root of both sides:
x = √28
Simplifying equation 2:
(x^2) = (y^2) - (y^2)
x = √(y^2) - √(y^2)
x = 0
This means x = 0, which indicates that the distance between the library and the park is 0 miles. In other words, the library and the park are the same point.
To summarize:
Distance between the library and the park = 0 miles.
Distance between the park and the football field = √28 miles.
Now we need to match these results with the given answer choices. Notice that none of the answer choices match the values we obtained. Therefore, none of the answer choices is correct for this specific problem.