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?
√22 miles; 2√10 miles
2√10 miles; √22 miles
6√2 miles; 6√11 miles
6√11 miles; 6√2 miles

I think it's C, but I'm not sure. Is that correct?

hmmm... I'm a bit confused. could you help me out with just how to set the equation up?

it is well known that the altitude of a right triangle is the geometric mean of the two parts of the hypotenuse.

See, for example,

http://jwilson.coe.uga.edu/emt668/emat6680.folders/brooks/6690stuff/righttriangle/rightday3.html

ohhh... so how would I use that information to help me get to the answer?

To solve this problem, we can use the Pythagorean Theorem. Let's break it down step by step:

1. Kristen's home is directly east of the park.
2. The football field is directly south of the park.
3. The library sits on the line formed between Kristen's home and the football field, where an altitude to the right triangle formed by her home, the park, and the football field can be drawn.
4. The library is 6 miles from Kristen's home.
5. The football field is 8 miles from the library.

To find the distance from the library to the park, we need to find the missing side of the right triangle formed by her home, the library, and the park.

Let's assume the distance from the park to the library is x miles. Now, we can set up the following equation:
6^2 + x^2 = (8 + x)^2

Simplifying this equation, we get:
36 + x^2 = 64 + 16x + x^2

Rearranging the terms, we have:
36 - 64 = 16x
-28 = 16x

Dividing both sides by 16, we get:
x = -28/16
x = -7/4

Based on the given information, the distance cannot be negative, so we need to discard this solution. It implies that our initial assumption was wrong.

Let's assume the distance from the library to the park is y miles. Now, we can set up a new equation:
6^2 + y^2 = (8 - y)^2

Simplifying this equation, we get:
36 + y^2 = 64 - 16y + y^2

Rearranging the terms, we have:
36 - 64 = -16y
-28 = -16y

Dividing both sides by -16, we get:
y = -28/-16
y = 7/4

Again, the distance cannot be negative, so we need to discard this solution as well.

Now, we need to consider the positive value for y, which gives us the distance from the library to the park. Therefore, the distance from the library to the park is √(22) miles.

To find the distance from the park to the football field, we can subtract the distance from the library to the park (√(22) miles) from the distance from the library to the football field (8 miles). This gives us:
8 - √(22) miles.

So, the distance from the park to the football field is 8 - √(22) miles.

Therefore, the correct answer is: √(22) miles; 8 - √(22) miles.

How far is the library from the park?

x^2 = 6*8

I don't see that as a choice.

??